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HERALD   OF   SCI""11 

Gift  Of 

IN    GROLIER    E.v.-Uri.-lSj« 

The  Dibner  Library 

of  the  History  of 
Science  and  Technology 


[    459    ] 

VIII.  A  Dynamical  Theory  of  the  Electromagnetic  Field.     By  J.  Clerk  Maxwell.  F.R.S. 

Received  October  27, — Read  December  8,  1864. 


(1)  The  most  obvious  mechanical  phenomenon  in  electrical  and  magnetical  experiments 
is  the  mutual  action  by  which  bodies  in  certain  states  set  each  other  in  motion  while 
still  at  a  sensible  distance  from  each  other.  The  first  step,  therefore,  in  reducing  these 
phenomena  into  scientific  form,  is  to  ascertain  the  magnitude  and  direction  of  the  force 
acting  between  the  bodies,  and  when  it  is  found  that  this  force  depends  in  a  certain 
way  upon  the  relative  position  of  the  bodies  and  on  their  electric  or  magnetic  condition, 
it  seems  at  first  sight  natural  to  explain  the  facts  by  assuming  the  existence  of  some- 
thing either  at  rest  or  in  motion  in  each  body,  constituting  its  electric  or  magnetic  state, 
and  capable  of  acting  at  a  distance  according  to  mathematical  laws. 

In  this  way  mathematical  theories  of  statical  electricity,  of  magnetism,  of  the  mecha- 
nical action  between  conductors  carrying  currents,  and  of  the  induction  of  currents  have 
been  formed.  In  these  theories  the  force  acting  between  the  two  bodies  is  treated  with 
reference  only  to  the  condition  of  the  bodies  and  their  relative  position,  and  without 
any  express  consideration  of  the  surrounding  medium. 

These  theories  assume,  more  or  less  explicitly,  the  existence  of  substances  the  parti- 
cles of  which  have  the  property  of  acting  on  one  another  at  a  distance  by  attraction 
or  repulsion.  The  most  complete  development  of  a  theory  of  this  kind  is  that  of 
M.  W.  Weber*,  who  has  made  the  same  theory  include  electrostatic  and  electromagnetic 

In  doing  so,  however,  he  has  found  it  necessary  to  assume  that  the  force  between 
two  electric  particles  depends  on  their  relative  velocity,  as  well  as  on  their  distance. 

This  theory,  as  developed  by  MM.  W.  Weber  and  C.  Neumann f,  is  exceedingly 
ingenious,  and  wonderfully  comprehensive  in  its  application  to  the  phenomena  of 
statical  electricity,  electromagnetic  attractions,  induction  of  currents  and  diamagnetic 
phenomena ;  and  it  comes  to  us  with  the  more  authority,  as  it  has  served  to  guide  the 
speculations  of  one  who  has  made  so  great  an  advance  in  the  practical  part  of  electric 
science,  both  by  introducing  a  consistent  system  of  units  in  electrical  measurement,  and 
by  actually  determining  electrical  quantities  with  an  accuracy  hitherto  unknown. 

*  Electrodynamische  Maassbestimmungen.     Leipzic  Trans,  vol.  i.  1849,andTAYLOR'sScientificMemoirs,  vol. v. 
art.  xiv. 

f  "  Explicare  tentatur  quomodo  fiat  ut  lucis  planum  polarizatiofcis  per  vires  electricas  vel  magneticas  decli- 
netur." — Halis  Saxonum,  1838. 



(2)  The  mechanical  difficulties,  however,  which  are  involved  in  the  assumption  of 
particles  acting  at  a  distance  with  forces  which  depend  on  their  velocities  are  such  as 
to  prevent  me  from  considering  this  theory  as  an  ultimate  one,  though  it  may  have  been, 
and  may  yet  be  useful  in  leading  to  the  coordination  of  phenomena. 

I  have  therefore  preferred  to  seek  an  explanation  of  the  fact  in  another  direction,  by 
supposing  them  to  be  produced  by  actions  which  go  on  in  the  surrounding  medium  as 
well  as  in  the  excited  bodies,  and  endeavouring  to  explain  the  action  between  distant 
bodies  without  assuming  the  existence  of  forces  capable  of  acting  directly  at  sensible 

(3)  The  theory  I  propose  may  therefore  be  called  a  theory  of  the  Electromagnetic  Field, 
because  it  has  to  do  with  the  space  in  the  neighbourhood  of  the  electric  or  magnetic  bodies, 
and  it  may  be  called  a  Dynamical  Theory,  because  it  assumes  that  in  that  space  there  is 
matter  in  motion,  by  which  the  observed  electromagnetic  phenomena  are  produced. 

(4)  The  electromagnetic  field  is  that  part  of  space  which  contains  and  surrounds 
bodies  in  electric  or  magnetic  conditions. 

It  may  be  filled  with  any  kind  of  matter,  or  we  may  endeavour  to  render  it  empty  of 
all  gross  matter,  as  in  the  case  of  Geissler's  tubes  and  other  so-called  vacua. 

There  is  always,  however,  enough  of  matter  left  to  receive  and  transmit  the  undulations 
of  light  and  heat,  and  it  is  because  the  transmission  of  these  radiations  is  not  greatly 
altered  when  transparent  bodies  of  measurable  density  are  substituted  for  the  so-called 
vacuum,  that  we  are  obliged  to  admit  that  the  undulations  are  those  of  an  sethereal 
substance,  and  not  of  the  gross  matter,  the  presence  of  which  merely  modifies  in  some 
way  the  motion  of  the  sether. 

We  have  therefore  some  reason  to  believe,  from  the  phenomena  of  light  and  heat, 
that  there  is  an  sethereal  medium  filling  space  and  permeating  bodies,  capable  of  being 
set  in  motion  and  of  transmitting  that  motion  from  one  part  to  another,  and  of  com- 
municating that  motion  to  gross  matter  so  as  to  heat  it  and  affect  it  in  various  ways. 

(5)  Now  the  energy  communicated  to  the  body  in  heating  it  must  have  formerly 
existed  in  the  moving  medium,  for  the  undulations  had  left  the  source  of  heat  some  time 
before  they  reached  the  body,  and  during  that  time  the  energy  must  have  been  half  in 
the  form  of  motion  of  the  medium  and  half  in  the  form  of  elastic  resilience.  From 
these  considerations  Professor  W.Thomson  has  argued*,  that  the  medium  must  have  a 
density  capable  of  comparison  with  that  of  gross  matter,  and  has  even  assigned  an  infe- 
rior limit  to  that  density. 

(6)  We  may  therefore  receive,  as  a  datum  derived  from  a  branch  of  science  inde- 
pendent of  that  with  which  we  have  to  deal,  the  existence  of  a  pervading  medium,  of 
small  but  real  density,  capable  of  being  set  in  motion,  and  of  transmitting  motion  from 
one  part  to  another  with  great,  but  not  infinite,  velocity. 

Hence  the  parts  of  this  medium  must  be  so  connected  that  the  motion  of  one  part 

*  "  On  the  Possible  Density  of  the  Luminiferous  Medium,  and  on  the  Mechanical  Value  of  a  Cubic  Mile  of 
Sunlight,"  Transactions  of  the  Royal  Society  of  Edinburgh  (1854),  p.  57. 



depends  in  some  way  on  the  motion  of  the  rest ;  and  at  the  same  time  these  connexions 
must  be  capable  of  a  certain  kind  of  elastic  yielding,  since  the  communication  of  motion 
is  not  instantaneous,  but  occupies  time. 

The  medium  is  therefore  capable  of  receiving  and  storing  up  two  kinds  of  energy, 
namely,  the  "  actual "  energy  depending  on  the  motions  of  its  parts,  and  "  potential " 
energy,  consisting  of  the  work  which  the  medium  will  do  in  recovering  from  displace- 
ment in  virtue  of  its  elasticity. 

The  propagation  of  undulations  consists  in  the  continual  transformation  of  one  of 
these  forms  of  energy  into  the  other  alternately,  and  at  any  instant  the  amount  of 
energy  in  the  whole  medium  is  equally  divided,  so  that  half  is  energy  of  motion,  and 
half  is  elastic  resilience. 

(7)  A  medium  having  such  a  constitution  may  be  capable  of  other  kinds  of  motion 
and  displacement  than  those  which  produce  the  phenomena  of  light  and  heat,  and  some 
of  these  may  be  of  such  a  kind  that  they  may  be  evidenced  to  our  senses  by  the  pheno- 
mena they  produce. 

(8)  Now  we  know  that  the  luminiferous  medium  is  in  certain  cases  acted  on  by 
magnetism;  for  Faeaday*  discovered  that  when  a  plane  polarized  ray  traverses  a  trans- 
parent diamagnetic  medium  in  the  direction  of  the  lines  of  magnetic  force  produced  by 
magnets  or  currents  in  the  neighbourhood,  the  plane  of  polarization  is  caused  to  rotate. 

This  rotation  is  always  in  the  direction  in  which  positive  electricity  must  be  carried 
round  the  diamagnetic  body  in  order  to  produce  the  actual  magnetization  of  the  field. 

M.  VEKDEif  has  since  discovered  that  if  a  paramagnetic  body,  such  as  solution  of 
perchloride  of  iron  in  ether,  be  substituted  for  the  diamagnetic  body,  the  rotation  is  in 
the  opposite  direction. 

Now  Professor  W.  ThomsoxJ  has  pointed  out  that  no  distribution  of  forces  acting 
between  the  parts  of  a  medium  whose  only  motion  is  that  of  the  luminous  vibrations,  is 
sufficient  to  account  for  the  phenomena,  but  that  we  must  admit  the  existence  of  a 
motion  in  the  medium  depending  on  the  magnetization,  in  addition  to  the  vibratory 
motion  which  constitutes  light. 

It  is  true  that  the  rotation  by  magnetism  of  the  plane  of  polarization  has  been 
observed  only  in  media  of  considerable  density ;  but  the  properties  of  the  magnetic  field 
are  not  so  much  altered  by  the  substitution  of  one  medium  for  another,  or  for  a  vacuum, 
as  to  allow  us  to  suppose  that  the  dense  medium  does  anything  more  than  merely  modify 
the  motion  of  the  ether.  We  have  therefore  warrantable  grounds  for  inquiring  whether 
there  may  not  be  a  motion  of  the  ethereal  medium  going  on  wherever  magnetic  effects 
are  observed,  and  we  have  some  reason  to  suppose  that  this  motion  is  one  of  rotation, 
having  the  direction  of  the  magnetic  force  as  its  axis. 

(9)  We  may  now  consider  another  phenomenon  observed  in  the  electromagnetic 

*  Experimental  Researches,  Series  19. 

t  Comptes  Rendus  (1856,  second  half  year,  p.  529,  and  1857,  first  half  year,  p.  1209). 

±  Proceedings  of  the  Royal  Society,  June  1856  and  June  1861. 



field.  When  a  body  is  moved  across  the  lines  of  magnetic  force  it  experiences  what  is 
called  an  electromotive  force ;  the  two  extremities  of  the  body  tend  to  become  oppo- 
sitely electrified,  and  an  electric  current  tends  to  flow  through  the  body.  When  the 
electromotive  force  is  sufficiently  powerful,  and  is  made  to  act  on  certain  compound 
bodies,  it  decomposes  them,  and  causes  one  of  their  components  to  pass  towards  one 
extremity  of  the  body,  and  the  other  in  the  opposite  direction. 

Here  we  have  evidence  of  a  force  causing  an  electric  current  in  spite  of  resist- 
ance; electrifying  the  extremities  of  a  body  in  opposite  ways,  a  condition  which  is 
sustained  only  by  the  action  of  the  electromotive  force,  and  which,  as  soon  as  that  force 
is  removed,  tends,  with  an  equal  and  opposite  force,  to  produce  a  counter  current  through 
the  body  and  to  restore  the  original  electrical  state  of  the  body ;  and  finally,  if  strong 
enough,  tearing  to  pieces  chemical  compounds  and  carrying  their  components  in  oppo- 
site directions,  while  their  natural  tendency  is  to  combine,  and  to  combine  with  a  force 
which  can  generate  an  electromotive  force  in  the  reverse  direction. 

This,  then,  is  a  force  acting  on  a  body  caused  by  its  motion  through  the  electro- 
magnetic field,  or  by  changes  occurring  in  that  field  itself;  and  the  effect  of  the  force  is 
either  to  produce  a  current  and  heat  the  body,  or  to  decompose  the  body,  or,  when  it 
can  do  neither,  to  put  the  body  in  a  state  of  electric  polarization, — a  state  of  constraint 
in  which  opposite  extremities  are  oppositely  electrified,  and  from  which  the  body  tends 
to  relieve  itself  as  soon  as  the  disturbing  force  is  removed. 

(10)  According  to  the  theory  which  I  propose  to  explain,  this  "electromotive  force" 
is  the  force  called  into  play  during  the  communication  of  motion  from  one  part  of  the 
medium  to  another,  and  it  is  by  means  of  this  force  that  the  motion  of  one  part  causes 
motion  in  another  part.  When  electromotive  force  acts  on  a  conducting  circuit,  it  pro- 
duces a  current,  which,  as  it  meets  with  resistance,  occasions  a  continual  transformation 
of  electrical  energy  into  heat,  which  is  incapable  of  being  restored  again  to  the  form  of 
electrical  energy  by  any  reversal  of  the  process. 

(11)  But  when  electromotive  force  acts  on  a  dielectric  it  produces  a  state  of  polari- 
zation of  its  parts  similar  in  distribution  to  the  polarity  of  the  parts  of  a  mass  of  iron 
under  the  influence  of  a  magnet,  and  like  the  magnetic  polarization,  capable  of  being 
described  as  a  state  in  which  every  particle  has  its  opposite  poles  in  opposite  con- 

In  a  dielectric  under  the  action  of  electromotive  force,  we  may  conceive  that  the 
electricity  in  each  molecule  is  so  displaced  that  one  side  is  rendered  positively  and  the 
other  negatively  electrical,  but  that  the  electricity  remains  entirely  connected  with  the 
molecule,  and  does  not  pass  from  one  molecule  to  another.  The  effect  of  this  action  on 
the  whole  dielectric  mass  is  to  produce  a  general  displacement  of  electricity  in  a  cer- 
tain direction.  This  displacement  does  not  amount  to  a  current,  because  when  it  has 
attained  to  a  certain  value  it  remains  constant,  but  it  is  the  commencement  of  a  current, 
and  its  variations  constitute  currents  in  the  positive  or  the  negative  direction  according 

*  Fakaday,  Exp.  Res.  Series  XL ;  Mossotti,  Mem.  della  Soc.  Italiana  (Modena),  vol.  xxiv.  part  2.  p.  49. 


as  the  displacement  is  increasing  or  decreasing.  In  the  interior  of  the  dielectric  there 
is  no  indication  of  electrification,  because  the  electrification  of  the  surface  of  any  molecule 
is  neutralized  by  the  opposite  electrification  of  the  surface  of  the  molecules  in  contact 
with  it ;  but  at  the  bounding  surface  of  the  dielectric,  where  the  electrification  is  not 
neutralized,  we  find  the  phenomena  which  indicate  positive  or  negative  electrification. 

The  relation  between  the  electromotive  force  and  the  amount  of  electric  displacement 
it  produces  depends  on  the  nature  of  the  dielectric,  the  same  electromotive  force  pro- 
ducing generally  a  greater  electric  displacement  in  solid  dielectrics,  such  as  glass  or 
sulphur,  than  in  air. 

(12)  Here,  then,  we  perceive  another  effect  of  electromotive  force,  namely,  electric 
displacement,  which  according  to  our  theory  is  a  kind  of  elastic  yielding  to  the  action 
of  the  force,  similar  to  that  which  takes  place  in  structures  and  machines  owing  to  the 
want  of  perfect  rigidity  of  the  connexions. 

(13)  The  practical  investigation  of  the  inductive  capacity  of  dielectrics  is  rendered 
difficult  on  account  of  two  disturbing  phenomena.  The  first  is  the  conductivity  of  the 
dielectric,  which,  though  in  many  cases  exceedingly  small,  is  not  altogether  insensible. 
The  second  is  the  phenomenon  called  electric  absorption*,  in  virtue  of  which,  when  the 
dielectric  is  exposed  to  electromotive  force,  the  electric  displacement  gradually  increases, 
and  when  the  electromotive  force  is  removed,  the  dielectric  does  not  instantly  return  to 
its  primitive  state,  but  only  discharges  a  portion  of  its  electrification,  and  when  left  to 
itself  gradually  acquires  electrification  on  its  surface,  as  the  interior  gradually  becomes 
depolarized.  Almost  all  solid  dielectrics  exhibit  this  phenomenon,  which  gives  rise  to 
the  residual  charge  in  the  Leyden  jar,  and  to  several  phenomena  of  electric  cables 
described  by  Mr.  F.  Jexkin  f . 

(14)  We  have  here  two  other  kinds  of  yielding  besides  the  yielding  of  the  perfect 
dielectric,  which  we  have  compared  to  a  perfectly  elastic  body.  The  yielding  due  to 
conductivity  may  be  compared  to  that  of  a  viscous  fluid  (that  is  to  say,  a  fluid  having 
great  internal  friction),  or  a  soft  solid  on  which  the  smallest  force  produces  a  permanent 
alteration  of  figure  increasing  with  the  time  during  which  the  force  acts.  The  yielding 
due  to  electric  absorption  may  be  compared  to  that  of  a  cellular  elastic  body  containing 
a  thick  fluid  in  its  cavities.  Such  a  body,  when  subjected  to  pressure,  is  compressed  by 
degrees  on  account  of  the  gradual  yielding  of  the  thick  fluid ;  and  when  the  pressure  is 
removed  it  does  not  at  once  recover  its  figure,  because  the  elasticity  of  the  substance  of 
the  body  has  gradually  to  overcome  the  tenacity  of  the  fluid  before  it  can  regain  com- 
plete equilibrium. 

Several  solid  bodies  in  which  no  such  structure  as  wre  have  supposed  can  be  found, 
seem  to  possess  a  mechanical  property  of  this  kind  $ ;  and  it  seems  probable  that  the 

*  Faraday,  Exp.  Ees.  1233-1250. 

t  Eeports  of  British  Association,  1859,  p.  248 ;  and  Eeport  of  Committee  of  Board  of  Trade  on  Submarine 
Cables,  pp.  136  &  464. 

X  As,  for  instance,  the  composition  of  glue,  treacle,  &c,  of  which  small  plastic  figures  are  made,  which  after 
being  distorted  gradually  recover  their  shape. 


same  substances,  if  dielectrics,  may  possess  the  analogous  electrical  property,  and  if 
magnetic,  may  have  corresponding  properties  relating  to  the  acquisition,  retention,  and 
loss  of  magnetic  polarity. 

(15)  It  appears  therefore  that  certain  phenomena  in  electricity  and  magnetism  lead 
to  the  same  conclusion  as  those  of  optics,  namely,  that  there  is  an  sethereal  medium 
pervading  all  bodies,  and  modified  only  in  degree  by  their  presence ;  that  the  parts  of 
this  medium  are  capable  of  being  set  in  motion  by  electric  currents  and  magnets ;  that 
this  motion  is  communicated  from  one  part  of  the  medium  to  another  by  forces  arising 
from  the  connexions  of  those  parts ;  that  under  the  action  of  these  forces  there  is  a 
certain  yielding  depending  on  the  elasticity  of  these  connexions ;  and  that  therefore 
energy  in  two  different  forms  may  exist  in  the  medium,  the  one  form  being  the  actual 
energy  of  motion  of  its  parts,  and  the  other  being  the  potential  energy  stored  up  in  the 
connexions,  in  virtue  of  their  elasticity. 

(1G)  Thus,  then,  we  are  led  to  the  conception  of  a  complicated  mechanism  capable 
of  a  vast  variety  of  motion,  but  at  the  same  time  so  connected  that  the  motion  of  one 
part  depends,  according  to  definite  relations,  on  the  motion  of  other  parts,  these  motions 
being  communicated  by  forces  arising  from  the  relative  displacement  of  the  connected 
parts,  in  virtue  of  their  elasticity.  Such  a  mechanism  must  be  subject  to  the  general 
laws  of  Dynamics,  and  we  ought  to  be  able  to  work  out  all  the  consequences  of  its 
motion,  provided  we  know  the  form  of  the  relation  between  the  motions  of  the  parts. 

(17)  We  know  that  when  an  electric  current  is  established  in  a  conducting  circuit, 
the  neighbouring  part  of  the  field  is  characterized  by  certain  magnetic  properties,  and 
that  if  two  circuits  are  in  the  field,  the  magnetic  properties  of  the  field  clue  to  the  two 
currents  are  combined.  Thus  each  part  of  the  field  is  in  connexion  with  both  currents, 
and  the  two  currents  are  put  in  connexion  with  each  ether  in  virtue  of  their  con- 
nexion with  the  magnetization  of  the  field.  The  first  result  of  this  connexion  that  I 
propose  to  examine,  is  the  induction  of  one  current  by  another,  and  by  the  motion  of 
conductors  in  the  field. 

The  second  result,  which  is  deduced  from  this,  is  the  mechanical  action  between  con- 
ductors carrying  currents.  The  phenomenon  of  the  induction  of  currents  has  been 
deduced  from  their  mechanical  action  by  Helmholtz*  and  Thomson  f.  I  have  followed 
the  reverse  order,  and  deduced  the  mechanical  action  from  the  laws  of  induction.  I 
have  then  described  experimental  methods  of  determining  the  quantities  L,  M,  N,  on 
which  these  phenomena  depend. 

(18)  I  then  apply  the  phenomena  of  induction  and  attraction  of  currents  to  the 
exploration  of  the  electromagnetic  field,  and  the  laying  down  systems  of  lines  of  mag- 
netic force  which  indicate  its  magnetic  properties.  By  exploring  the  same  field  with  a 
magnet,  I  show  the  distribution  of  its  equipotential  magnetic  surfaces,  cutting  the  lines 
of  force  at  right  angles. 

*  "Conservation  of  Force,"  Physical  Society  of  Berlin,  1847;  and  Taylor's  Scientific  Memoirs,  1853, 
p.  114. 

t  Reports  of  the  British  Association,  1848;  Philosophical  Magazine,  Dec.  1851. 


In  order  to  bring  these  results  within  the  power  of  symbolical  calculation,  I  then 
express  them  in  the  form  of  the  General  Equations  of  the  Electromagnetic  Field. 
These  equations  express — 

(A)  The  relation  between  electric  displacement,  true  conduction,  and  the  total 
current,  compounded  of  both. 

(B)  The  relation  between  the  lines  of  magnetic  force  and  the  inductive  coefficients  of 
a  circuit,  as  already  deduced  from  the  laws  of  induction. 

(C)  The  relation  between  the  strength  of  a  current  and  its  magnetic  effects,  according 
to  the  electromagnetic  system  of  measurement. 

(D)  The  value  of  the  electromotive  force  in  a  body,  as  arising  from  the  motion  of  the 
body  in  the  field,  the  alteration  of  the  field  itself,  and  the  variation  of  electric 
potential  from  one  part  of  the  field  to  another. 

(E)  The  relation  between  electric  displacement,  and  the  electromotive  force  which 
produces  it. 

(F)  The  relation  between  an  electric  current,  and  the  electromotive  force  which  pro- 

duces it. 

(G)  The  relation  between  the  amount  of  free  electricity  at  any  point,  and  the  electric 
displacements  in  the  neighbourhood. 

(H)  The  relation  between  the  increase  or  diminution  of  free  electricity  and  the  elec- 
tric currents  in  the  neighbourhood. 
There  are  twenty  of  these  equations  in  all,  involving  twenty  variable  quantities. 

(19)  I  then  express  in  terms  of  these  quantities  the  intrinsic  energy  of  the  Electro- 
magnetic Field  as  depending  partly  on  its  magnetic  and  partly  on  its  electric  polariza- 
tion at  every  point. 

From  this  I  determine  the  mechanical  force  acting,  1st,  on  a  moveable  conductor 
carrying  an  electric  current ;  2ndly,  on  a  magnetic  pole ;   3rdly,  on  an  electrified  body. 

The  last  result,  namely,  the  mechanical  force  acting  on  an  electrified  body,  gives  rise 
to  an  independent  method  of  electrical  measurement  founded  on  its  electrostatic  effects. 
The  relation  between  the  units  employed  in  the  two  methods  is  shown  to  depend  on 
what  I  have  called  the  "  electric  elasticity"  of  the  medium,  and  to  be  a  velocity,  which 
has  been  experimentally  determined  by  MM.  Weber  and  Kohlrausch. 

I  then  show  how  to  calculate  the  electrostatic  capacity  of  a  condenser,  and  the 
specific  inductive  capacity  of  a  dielectric. 

The  case  of  a  condenser  composed  of  parallel  layers  of  substances  of  different  electric 
resistances  and  inductive  capacities  is  next  examined,  and  it  is  shown  that  the  pheno- 
menon called  electric  absorption  will  generally  occur,  that  is,  the  condenser,  when 
suddenly  discharged,  will  after  a  short  time  show  signs  of  a  residual  charge. 

(20)  The  general  equations  are  next  applied  to  the  case  of  a  magnetic  disturbance 
propagated  through  a  non-conducting  field,  and  it  is  shown  that  the  only  disturbances 
which  can  be  so  propagated  are  those  which  are  transverse  to  the  direction  of  propaga- 
tion, and  that  the  velocity  of  propagation  is  the  velocity  v,  found  from  experiments  such 


as  those  of  Weber,  which  expresses  the  number  of  electrostatic  units  of  electricity 
which  are  contained  in  one  electromagnetic  unit. 

This  velocity  is  so  nearly  that  of  light,  that  it  seems  we  have  strong  reason  to  con- 
clude that  light  itself  (including  radiant  heat,  and  other  radiations  if  any)  is  an  electro- 
magnetic disturbance  in  the  form  of  waves  propagated  through  the  electromagnetic  field 
according  to  electromagnetic  laws.  If  so,  the  agreement  between  the  elasticity  of  the 
medium  as  calculated  from  the  rapid  alternations  of  luminous  vibrations,  and  as  found 
by  the  slow  processes  of  electrical  experiments,  shows  how  perfect  and  regular  the 
elastic  properties  of  the  medium  must  be  when  not  encumbered  with  any  matter  denser 
than  air.  If  the  same  character  of  the  elasticity  is  retained  in  dense  transparent  bodies, 
it  appears  that  the  square  of  the  index  of  refraction  is  equal  to  the  product  of  the 
specific  dielectric  capacity  and  the  specific  magnetic  capacity.  Conducting  media  are 
shown  to  absorb  such  radiations  rapidly,  and  therefore  to  be  generally  opaque. 

The  conception  of  the  propagation  of  transverse  magnetic  disturbances  to  the  exclu- 
sion of  normal  ones  is  distinctly  set  forth  by  Professor  Faraday*  in  his  "Thoughts  on 
Ray  Vibrations."  The  electromagnetic  theory  of  light,  as  proposed  by  him,  is  the  same 
in  substance  as  that  which  I  have  begun  to  develope  in  this  paper,  except  that  in  1846 
there  were  no  data  to  calculate  the  velocity  of  propagation. 

(21)  The  general  equations  are  then  applied  to  the  calculation  of  the  coefficients  of 
mutual  induction  of  two  circular  currents  and  the  coefficient  of  self-induction  in  a  coil. 
The  want  of  uniformity  of  the  current  in  the  different  parts  of  the  section  of  a  wire  at 
the  commencement  of  the  current  is  investigated,  I  believe  for  the  first  time,  and  the 
consequent  correction  of  the  coefficient  of  self-induction  is  found. 

These  results  are  applied  to  the  calculation  of  the  self-induction  of  the  coil  used  in 
the  experiments  of  the  Committee  of  the  British  Association  on  Standards  of  Electric 
Resistance,  and  the  value  compared  with  that  deduced  from  the  experiments. 

Electromagnetic  Momentum  of  a  Current. 

(22)  We  may  begin  by  considering  the  state  of  the  field  in  the  neighbourhood  of  an 
electric  current.  We  know  that  magnetic  forces  are  excited  in  the  field,  their  direction 
and  magnitude  depending  according  to  known  laws  upon  the  form  of  the  conductor 
carrying  the  current.  When  the  strength  of  the  current  is  increased,  all  the  magnetic 
effects  are  increased  in  the  same  proportion.  Now,  if  the  magnetic  state  of  the  field 
depends  on  motions  of  the  medium,  a  certain  force  must  be  exerted  in  order  to  increase 
or  diminish  these  motions,  and  when  the  motions  are  excited  they  continue,  so  that  the 
effect  of  the  connexion  between  the  current  and  the  electromagnetic  field  surrounding 
it,  is  to  endow  the  current  with  a  kind  of  momentum,  just  as  the  connexion  between 
the  driving-point  of  a  machine  and  a  fly-wheel  endows  the  driving-point  with  an  addi- 

*  Philosophical  Magazine,  May  1846,  or  Experimental  Researches,  hi.  p.  447. 


tional  momentum,  which  may  be  called  the  momentum  of  the  fly-wheel  reduced  to 
the  driving-point.  The  unbalanced  force  acting  on  the  driving-point  increases  this 
momentum,  and  is  measured  by  the  rate  of  its  increase. 

In  the  case  of  electric  currents,  the  resistance  to  sudden  increase  or  diminution  of 
strength  produces  effects  exactly  like  those  of  momentum,  but  the  amount  of  this  mo- 
mentum depends  on  the  shape  of  the  conductor  and  the  relative  position  of  its  different 

Mutual  Action  of  two  Currents. 

(23)  If  there  are  two  electric  currents  in  the  field,  the  magnetic  force  at  any  point  is 
that  compounded  of  the  forces  due  to  each  current  separately,  and  since  the  two  currents 
are  in  connexion  with  every  point  of  the  field,  they  will  be  in  connexion  with  each  other, 
so  that  any  increase  or  diminution  of  the  one  will  produce  a  force  acting  with  or  con- 
trary to  the  other. 

Dynamical  Illustration  of  Reduced  Momentum. 

(24)  As  a  dynamical  illustration,  let  us  suppose  a  body  C  so  connected  with  two 
independent  driving-points  A  and  B  that  its  velocity  is  j)  times  that  of  A  together  with 
q  times  that  of  B.  Let  u  be  the  velocity  of  A,  v  that  of  B,  and  w  that  of  C,  and  let  &r, 
ty,  h  be  their  simultaneous  displacements,  then  by  the  general  equation  of  dynamics*, 

where  X  and  Y  are  the  forces  acting  at  A  and  B. 


dw        du        dv 


Substituting,  and  remembering  that  &r  and  hj  are  independent, 




We  may  call  Cj)2u-\-Cj)qv  the  momentum  of  C  referred  to  A,  and  Cpqu+Cq^v  its 
momentum  referred  to  B ;  then  we  may  say  that  the  effect  of  the  force  X  is  to  increase  the 
momentum  of  C  referred  to  A,  and  that  of  Y  to  increase  its  momentum  referred  to  B. 

If  there  are  many  bodies  connected  with  A  and  B  in  a  similar  way  but  with  different 
values  of  p  and  q,  we  may  treat  the  question  in  the  same  way  by  assuming 

L=2(Cp2),    M=2(Q>£),    andN=2(Cf), 

*  Lagbange,  Mec.  Anal.  ii.  2.  §  5. 


where  the  summation  is  extended  to  all  the  bodies  with  their  proper  values  of  C,  p,  and  q. 
Then  the  momentum  of  the  system  referred  to  A  is 

and  referred  to  B, 
and  we  shall  have 



Y  =  jt(Mu+Xv), 

where  X  and  Y  are  the  external  forces  acting  on  A  and  B. 

(25)  To  make  the  illustration  more  complete  we  have  only  to  suppose  that  the 
motion  of  A  is  resisted  by  a  force  proportional  to  its  velocity,  which  we  may  call  Ru, 
and  that  of  B  by  a  similar  force,  which  we  may  call  Sv,  R  and  S  being  coefficients  of 
resistance.     Then  if  |  and  q  are  the  forces  on  A  and  B 

%=x+Uu=~Ru+jt(Lu+m), ] 



If  the  velocity  of  A  be  increased  at  the  rate  -^;  then  in  order  to  prevent  B  from  moving 

a  force,  ?;=  -j-.(Mm)  must  be  applied  to  it. 

This  effect  on  B,  due  to  an  increase  of  the  velocity  of  A,  corresponds  to  the  electro- 
motive force  on  one  circuit  arising  from  an  increase  in  the  strength  of  a  neighbouring 

This  dynamical  illustration  is  to  be  considered  merely  as  assisting  the  reader  to  under- 
stand what  is  meant  in  mechanics  by  Reduced  Momentum.  The  facts  of  the  induction 
of  currents  as  depending  on  the  variations  of  the  quantity  called  Electromagnetic  Mo- 
mentum, or  Electrotonic  State,  rest  on  the  experiments  of  Faraday*,  Felici^,  &c. 

Coefficients  of  Induction  for  Two  Circuits. 

(26)  In  the  electromagnetic  field  the  values  of  L,  M,  N  depend  on  the  distribution 
of  the  magnetic  effects  due  to  the  two  circuits,  and  this  distribution  depends  only  on 
the  form  and  relative  position  of  the  circuits.  Hence  L,  M,  N  are  quantities  depending 
on  the  form  and  relative  position  of  the  circuits,  and  are  subject  to  variation  with  the 
motion  of  the  conductors.  It  will  be  presently  seen  that  L,  M,  N  are  geometrical 
quantities  of  the  nature  of  lines,  that  is,  of  one  dimension  in  space ;  L  depends  on  the 
form  of  the  first  conductor,  which  we  shall  call  A,  N  on  that  of  the  second,  which  we 
shall  call  B,  and  M  on  the  relative  position  of  A  and  B. 

(27)  Let  |  be  the  electromotive  force  acting  on  A,  x  the  strength  of  the  current,  and 
*  Experimental  Researches,  Series  I.,  IX.  t  Annales  de  Chimie,  ser.  3.  xxxiv.  (1852)  p.  64. 


E  the  resistance,  then  Rr  will  be  the  resisting  force.  In  steady  currents  the  electro- 
motive force  just  balances  the  resisting  force,  but  in  variable  currents  the  resultant 
force  ;=Rr  is  expended  in  increasing  the  "electromagnetic  momentum,"  using  the 
word  momentum  merely  to  express  that  which  is  generated  by  a  force  acting  during  a 
time,  that  is,  a  velocity  existing  in  a  body. 

In  the  case  of  electric  currents,  the  force  in  action  is  not  ordinary  mechanical  force,  at 
least  we  are  not  as  yet  able  to  measure  it  as  common  force,  but  we  call  it  electromotive 
force,  and  the  body  moved  is  not  merely  the  electricity  in  the  conductor,  but  something 
outside  the  conductor,  and  capable  of  being  affected  by  other  conductors  in  the  neighbour- 
hood carrying  currents.  In  this  it  resembles  rather  the  reduced  momentum  of  a  driving- 
point  of  a  machine  as  influenced  by  its  mechanical  connexions,  than  that  of  a  simple 
moving  body  like  a  cannon  ball,  or  water  in  a  tube. 

Electromagnetic  Relations  of  two  Conducting  Circuits. 

(28.)  In  the  case  of  two  conducting  circuits,  A  and  B,  we  shall  assume  that  the 
electromagnetic  momentum  belonging  to  A  is 

and  that  belonging  to  B, 

M#  +  %, 

where  L,  M,  N  correspond  to  the  same  quantities  in  the  dynamical  illustration,  except 
that  they  are  supposed  to  be  capable  of  variation  when  the  conductors  A  or  B  are 

Then  the  equation  of  the  current  x  in  A  will  be 

g=R#+|(Ltf+3V%), (4) 

and  that  of  y  in  B 

,=S^  +  |(M^+%), (5) 

where  f  and  r,  are  the  electromotive  forces,  x  and  y  the  currents,  and  R  and  S  the 
resistances  in  A  and  B  respectively. 

Induction  of  one  Current  by  another. 

(29)  Case  1st.  Let  there  be  no  electromotive  force  on  B,  except  that  which  arises 
from  the  action  of  A,  and  let  the  current  of  A  increase  from  0  to  the  value  x,  then 



V—  f  „,7/—  _ 

cl  M 

Y=l  ydt=—-^x, 

that  is,  a  quantity  of  electricity  Y,  being  the  total  induced  current,  will  flow  through  B 
when  x  rises  from  0  to  x.     This  is  induction  by  variation  of  the  current  in  the  primary 

3s  2 


conductor.  When  M  is  positive,  the  induced  current  due  to  increase  of  the  primary 
current  is  negative. 

Induction  by  Motion  of  Conductor. 

(30)  Case  2nd.  Let  x  remain  constant,  and  let  M  change  from  M  to  M',  then 

v_      M~M 
i_—      s     x, 

so  that  if  M  is  increased,  which  it  will  be  by  the  primary  and  secondary  circuits 
approaching  each  other,  there  will  be  a  negative  induced  current,  the  total  quantity  of 
electricity  passed  through  B  being  Y. 

This  is  induction  by  the  relative  motion  of  the  primary  and  secondary  conductors. 

Equation  of  Work  and  Energy. 

(31)  To  form  the  equation  between  work  done  and  energy  produced,  multiply  (1)  by 
x  and  (2)  by  y,  and  add 

^+rly=T^+Sf+x^(Lx  +  My)+y^lx^y) (8) 

Here  \x  is  the  work  clone  in  unit  of  time  by  the  electromotive  force  %  acting  on  the 
current  x  and  maintaining  it,  and  ry  is  the  work  done  by  the  electromotive  force  q. 
Hence  the  left-hand  side  of  the  equation  represents  the  work  done  by  the  electromotive 
forces  in  unit  of  time. 

Heat  produced  by  the  Current. 

(32)  On  the  other  side  of  the  equation  we  have,  first, 

E^  +  S/=H, (9) 

which  represents  the  work  done  in  overcoming  the  resistance  of  the  circuits  in  unit  of 
time.  This  is  converted  into  heat.  The  remaining  terms  represent  work  not  converted 
into  heat.     They  may  be  written 


Intrinsic  Energy  of  the  Currents. 

(33)  If  L,  M,  N  are  constant,  the  whole  work  of  the  electromotive  forces  which  is 
not  spent  against  resistance  will  be  devoted  to  the  development  of  the  currents.  The 
whole  intrinsic  energy  of  the  currents  is  therefore 

iL^+M^+i%2=E (10) 

This  energy  exists  in  a  form  imperceptible  to  our  senses,  probably  as  actual  motion,  the 
seat  of  this  motion  being  not  merely  the  conducting  circuits,  but  the  space  surrounding 


Mechanical  Action  between  Conductors. 
(34)  The  remaining  terms, 

if^+f  **+if*'=w (ii) 

represent  the  work  done  in  unit  of  time  arising  from  the  variations  of  L,  M,  and  N,  or, 
what  is  the  same  thing,  alterations  in  the  form  and  position  of  the  conducting  circuits 
A  and  B. 

Now  if  work  is  done  when  a  body  is  moved,  it  must  arise  from  ordinary  mechanical 
force  acting  on  the  body  while  it  is  moved.  Hence  this  part  of  the  expression  shows 
that  there  is  a  mechanical  force  urging  every  part  of  the  conductors  themselves  in  that 
direction  in  which  L,  M,  and  N  will  be  most  increased. 

The  existence  of  the  electromagnetic  force  between  conductors  carrying  currents  is 
therefore  a  direct  consequence  of  the  joint  and  independent  action  of  each  current  on 
the  electromagnetic  field.  If  A  and  B  are  allowed  to  approach  a  distance  ds,  so  as  to 
increase  M  from  M  to  M'  while  the  currents  are  x  and  y,  then  the  work  done  will  be 

and  the  force  in  the  direction  of  ds  will  be 

£» (12) 

and  this  will  be  an  attraction  if  x  and  y  are  of  the  same  sign,  and  if  M  is  increased  as 
A  and  B  approach. 

It  appears,  therefore,  that  if  we  admit  that  the  unresisted  part  of  electromotive  force 
goes  on  as  long  as  it  acts,  generating  a  self-persistent  state  of  the  current,  which 
we  may  call  (from  mechanical  analogy)  its  electromagnetic  momentum,  and  that  this 
momentum  depends  on  circumstances  external  to  the  conductor,  then  both  induction  of 
currents  and  electromagnetic  attractions  may  be  proved  by  mechanical  reasoning. 

What  I  have  called  electromagnetic  momentum  is  the  same  quantity  which  is  called 
by  Faraday*  the  electrotonic  state  of  the  circuit,  every  change  of  which  involves  the 
action  of  an  electromotive  force,  just  as  change  of  momentum  involves  the  action  of 
mechanical  force. 

If,  therefore,  the  phenomena  described  by  Faraday  in  the  Ninth  Series  of  his  Expe- 
rimental Researches  were  the  only  known  facts  about  electric  currents,  the  laws  of 
Ampere  relating  to  the  attraction  of  conductors  carrying  currents,  as  well  as  those 
of  Faraday  about  the  mutual  induction  of  currents,  might  be  deduced  by  mechanical 

In  order  to  bring  these  results  within  the  range  of  experimental  verification,  I  shall 
next  investigate  the  case  of  a  single  current,  of  two  currents,  and  of  the  six  currents 
in  the  electric  balance,  so  as  to  enable  the  experimenter  to  determine  the  values  of 
L5  M,  N. 

*  Expeiimental  Researches,  Series  I.  60,  &c. 


Case  of  a  single  Circuit. 

(35)  The  equation  of  the  current  x  in  a  circuit  whose  resistance  is  R,  and  whose 
coefficient  of  self-induction  is  L,  acted  on  by  an  external  electromotive  force  g,  is 

5-H*HgI* (13) 

When  5  is  constant,  the  solution  is  of  the  form 

x=b-\-(a— b)e   L  , 

where  a  is  the  value  of  the  current  at  the  commencement,  and  b  is  its  final  value. 
The  total  quantity  of  electricity  which  passes  in  time  t,  where  t  is  great,  is 


xclt=bt-\-(a-b)~ (14) 

The  value  of  the  integral  of  x1  with  respect  to  the  time  is 

£w*dt=bH+(a-b)^*±?} (15) 

The  actual  current  changes  gradually  from  the  initial  value  a  to  the  final  value  b,  but 
the  values  of  the  integrals  of  x  and  x2  are  the  same  as  if  a  steady  current  of  intensity 

\(a-\-b)  were  to  flow  for  a  time  2-^-,  and  were  then  succeeded  by  the  steady  current  b. 


The  time  2  —  is  generally  so  minute  a  fraction  of  a  second,  that  the  effects  on  the  galvano- 

meter  and  dynamometer  may  be  calculated  as  if  the  impulse  were  instantaneous. 

If  the  circuit  consists  of  a  battery  and  a  coil,  then,  when  the  circuit  is  first  completed, 
the  effects  are  the  same  as  if  the  current  had  only  half  its  final  strength  during  the  time 

2  — .     This  diminution  of  the  current,  due  to  induction,  is  sometimes  called  the  counter- 


(36)  If  an  additional  resistance  r  is  suddenly  thrown  into  the  circuit,  as  by  breaking 

contact,  so  as  to  force  the  current  to  pass  through  a  thin  wire  of  resistance  r,  then  the 

t  t 

original  current  is  # =^,  and  the  final  current  is  b~ 

R1  R  +  r 

The  current  of  induction  is  then  i|„  n  -   ,  and  continues  for  a  time  2- — .     This 

R(R  +  r)  R  +  ?- 

current  is  greater  than  that  which  the  battery  can  maintain  in  the  two  wires  E  and  r, 

and  may  be  sufficient  to  ignite  the  thin  wire  r. 

When  contact  is  broken  by  separating  the  wires  in  air,  this  additional  resistance  is 

given  by  the  interposed  air,  and  since  the  electromotive  force  across  the  new  resistance 

is  very  great,  a  spark  will  be  forced  across. 


If  the  electromotive  force  is  of  the  form  E  sinpt,  as  in  the  case  of  a  coil  revolving  in 
a  magnetic  field,  then 

x=— sm(])t— a), 

where  %2=~R2-\-'L2])2,  and  tana=^- 

Case  of  two  Circuits. 

(37)  Let  E  be  the  primary  circuit  and  S  the  secondary  circuit,  then  we  have  a  case 
similar  to  that  of  the  induction  coil. 

The  equations  of  currents  are  those  marked  A  and  B,  and  we  may  here  assume 
L,  M,  N  as  constant  because  there  is  no  motion  of  the  conductors.  The  equations 
then  become 

R*+L  Tt+^{t=t, 


To  find  the  total  quantity  of  electricity  which  passes,  we  have  only  to  integrate  these 
equations  with  respect  to  t;  then  if  #0,  y0  be  the  strengths  of  the  currents  at  time  0, 
and  xx ,  yx  at  time  t,  and  if  X,  Y  be  the  quantities  of  electricity  passed  through  each 
circuit  during  time  t, 

X=R^  +  L(^-"T'HM(2/o-y.)}, 

When  the  circuit  R  is  completed,  then  the  total  currents  up  to  time  t,  when  t  is 
great,  are  found  by  making 


x0=0,     Xy=%,     ?/o=0,    ^=0; 



X=*,(*-|),     Y=-|^ .     (15*) 

The  value  of  the  total  counter-current  in  R  is  therefore  independent  of  the  secondary 
circuit,  and  the  induction  current  in  the  secondary  circuit  depends  only  on  M,  the 
coeflicient  of  induction  between  the  coils,  S  the  resistance  of  the  secondary  coil,  and 
xx  the  final  strength  of  the  current  in  R. 

When  the  electromotive  force  |  ceases  to  act,  there  is  an  extra  current  in  the  pri- 
mary circuit,  and  a  positive  induced  current  in  the  secondary  circuit,  whose  values  are 
equal  and  opposite  to  those  produced  on  making  contact. 

(38)  All  questions  relating  to  the  total  quantity  of  transient  currents,  as  measured 
by  the  impulse  given  to  the  magnet  of  the  galvanometer,  may  be  solved  in  this  way 
without  the  necessity  of  a  complete  solution  of  the  equations.     The  heating  effect  of 


the  current,  and  the  impulse  it  gives  to  the  suspended  coil  of  Weber's  dynamometer, 
depend  on  the  square  of  the  current  at  every  instant  during  the  short  time  it  lasts. 
Hence  we  must  obtain  the  solution  of  the  equations,  and  from  the  solution  we  may  find 
the  effects  both  on  the  galvanometer  and  dynamometer ;  and  we  may  then  make  use  of 
the  method  of  Weber  for  estimating  the  intensity  and  duration  of  a  current  uniform 
while  it  lasts  which  would  produce  the  same  effects. 
(39)  Let  wn  n2  be  the  roots  of  the  equation 

(LN-M>2-|-(RN+LS>+RS=0,       (16) 

and  let  the  primary  coil  be  acted  on  by  a  constant  electromotive  force  He,  so  that  c  is 
the  constant  current  it  could  maintain ;  then  the  complete  solution  of  the  equations  for 
making  contact  is 

^^((-+N)^-f-+N)^+S^l       ....     (17) 

y=T^<**-'*> (18) 

From  these  we  obtain  for  calculating  the  impulse  on  the  dynamometer, 

^dt^At~\\-\-^-\     (i9) 

2R     2RN  +  LSj' 
fcAw-cU M2R (20) 

The  effects  of  the  current  in  the  secondary  coil  on  the  galvanometer  and  dynamometer 
are  the  same  as  those  of  a  uniform  current 

_i        MR 

_2<?RN  +  LS 
for  a  time 

(40)  The  equation  between  work  and  energy  may  be  easily  verified.     The  work  done 
by  the  electromotive  force  is 


Work  done  in  overcoming  resistance  and  producing  heat, 


Energy  remaining  in  the  system, 


(41)  If  the  circuit  R  is  suddenly  and  completely  interrupted  while  carrying  a  current 
c,  then  the  equation  of  the  current  in  the  secondary  coil  would  be 

M   _ 


y=c-e  »  . 
This  current  begins  with  a  value  c  -^ ,  and  gradually  disappears. 



The  total  quantity  of  electricity  is  c-^- ,  and  the  value  of  §y2dt  is  c2 

The  effects  on  the  galvanometer  and  dynamometer  are  equal  to  those  of  a  uniform 


current  A  c  ^  for  a  time  2  - 

The  heating  effect  is  therefore  greater  than  that  of  the  current  on  making  contact. 
(42)  If  an  electromotive  force  of  the  form  £=E  cos  pt  acts  on  the  circuit  R,  then  if 
the  circuit  S  is  removed,  the  value  of  x  will  be 



w=  j  sm(pt-x), 

tan  a= 



The  effect  of  the  presence  of  the  circuit  S  in  the  neighbourhood  is  to  alter  the  value 
of  A  and  a,  to  that  which  they  would  be  if  R  become 



and  L  became 

L—  jy 




Hence  the  effect  of  the  presence  of  the  circuit  S  is  to  increase  the  apparent  resistance  and 
diminish  the  apparent  self-induction  of  the  circuit  R. 

On  the  Determination  of  Coefficients  of  Induction  by  the  Electric  Balance. 

(43)  The  electric  balance  consists  of  six  con- 
ductors joining  four  points,  A  C  D  E,  two  and  two. 
One  pair,  A  C,  of  these  points  is  connected  through 
the  battery  B.  The  opposite  pair,  D  E,  is  connected 
through  the  galvanometer  G.  Then  if  the  resistances 
of  the  four  remaining  conductors  are  represented  by 
P,  Q,  R,  S,  and  the  currents  in  them  by  #,  %—z,  y, 
and  y-\-z,  the  current  through  G  will  be  z.  Let  the 
potentials  at  the  four  points  be  A,  C,  D,  E.  Then  the  conditions  of  steady  currents  may 
be  found  from  the  equations 

Po?=A-D  Q(x-z)=D-C, 

R^=A-E  S(y+z)=E-C,  (21) 

Gz=D-E  B(#+^)=-A+C+F. 

Solving  these  equations  for  z,  we  find 

4p+Q+E+H+B(f+h)(q+I)+g(p+4)(s+I)+pSs(P+Q+E+S)}=F(^-5r)-    (22) 
mdccclxv.  3  t 



























In  this  expression  F  is  the  electromotive  force  of  the  battery,  z  the  current  through 
the  galvanometer  when  it  has  become  steady.  P,  Q,  Pi,  S  the  resistances  in  the  four 
arms.     B  that  of  the  battery  and  electrodes,  and  G  that  of  the  galvanometer. 

(44)  If  PS  =  QR,  then  z  =  0,  and  there  will  be  no  steady  current,  but  a  transient 
current  through  the  galvanometer  may  be  produced  on  making  or  breaking  circuit  on 
account  of  induction,  and  the  indications  of  the  galvanometer  may  be  used  to  determine 
the  coefficients  of  induction,  provided  we  understand  the  actions  which  take  place. 

We  shall  suppose  PS=QR,  so  that  the  current  z  vanishes  when  sufficient  time  is 
allowed,  and 

rfPJ-f»-wni-uSt F(P  +  Q)(R  +  S) 

a:tr-t-u/— yi-K-i-DJ—  (p  +  Q){£+s)+B(P+Q)(R+S)' 

Let  the  induction  coefficients  between  P,  Q,  R  S,  be 
given  by  the  following  Table,  the  coefficient  of  induction 
of  P  on  itself  being  p,  between  P  and  Q,  h,  and  so  on. 

Let  g  be  the  coefficient  of  induction  of  the  galvanometer 
on  itself,  and  let  it  be  out  of  the  reach  of  the  inductive 
influence  of  P,  Q,  R,  S  (as  it  must  be  in  order  to  avoid 

direct  action  of  P,  Q,  R,  S  on  the  needle).  Let  X,  Y,  Z  be  the  integrals  of  x,  y,  z 
with  respect  to  t.  At  making  contact  x,  y,  z  are  zero.  After  a  time  z  disappears,  and 
x  and  y  reach  constant  values.     The  equations  for  each  conductor  will  therefore  be 

PX  +(P+h  )x+{k  +1  )y=SAdt-$Ddt, 

Q(X— Z)  +  (h+q  )x+{m+n)y  =$Ddt—$Cdt, 

RY  +(k+m)x+(r  +o)y=]\dt-$Edt,  [.  (24) 

S(Y+Z)  +(l  +n  )x+(o  +s)y=$Edt-$Cdt, 


Solving  these  equations  for  Z,  we  find 

=  -FP^{p-5-l+l+'<p-Q)+*(s-p)+<ri+5)-'»(p+s)       ■  (25) 

+»(M)+4-*)}  ! 

(45)  Now  let  the  deflection  of  the  galvanometer  by  the  instantaneous  current  whose 
intensity  is  Z  be  a. 

Let  the  permanent  deflection  produced  by  making  the  ratio  of  PS  to  QR,  %  instead  v#f 
unity,  be  0. 

Also  let  the  time  of  vibration  of  the  galvanometer  needle  from  rest  to  rest  be  T. 


Then  calling  the  quantity 

P     Q 


we  find  r,     n  ■    i     m 

Z 2sinA«  l_     t  (27) 

z         tanfl     ir      1  —  g 

In  determining  r  by  experiment,  it  is  best  to  make  the  alteration  of  resistance  in  one 
of  the  arms  by  means  of  the  arrangement  described  by  Mr.  Jenkin  in  the  Keport  of  the 
British  Association  for  1863,  by  which  any  value  of  §  from  1  to  1*01  can  be  accurately 

We  observe  (a)  the  greatest  deflection  due  to  the  impulse  of  induction  when  the 
galvanometer  is  in  circuit,  when  the  connexions  are  made,  and  when  the  resistances  are 
so  adjusted  as  to  give  no  permanent  current. 

We  then  observe  (j3)  the  greatest  deflection  produced  by  the  permanent  current  when 
the  resistance  of  one  of  the  arms  is  increased  in  the  ratio  of  1  to  g>,  the  galvanometer 
not  being  in  circuit  till  a  little  while  after  the  connexion  is  made  with  the  battery. 

In  order  to  eliminate  the  effects  of  resistance  of  the  air,  it  is  best  to  vary  §  till  j8  =  2a 
nearly;  then  .  .  .    , 

r=T^l-f)2^ (28) 

tt\        s'  tan  \p 

If  all  the  arms  of  the  balance  except  P  consist  of  resistance  coils  of  very  fine  wire  of 
no  great  length  and  doubled  before  being  coiled,  the  induction  coefficients  belonging  to 

these  coils  will  be  insensible,  and  r  will  be  reduced  to  ^.  The  electric  balance  there- 
fore affords  the  means  of  measuring  the  self-induction  of  any  circuit  whose  resistance  is 

(46)  It  may  also  be  used  to  determine  the  coefficient  of  induction  between  two 
circuits,  as  for  instance,  that  between  P  and  S  which  we  have  called  m ;  but  it  would  be 
more  convenient  to  measure  this  by  directly  measuring  the  current,  as  in  (37),  without 

using  the  balance.     We  may  also  ascertain  the  equality  of  ^  and  -3-  by  there  being  no 

current  of  induction,  and  thus,  when  we  know  the  value  of  p,  we  may  determine  that  of 
q  by  a  more  perfect  method  than  the  comparison  of  deflections. 

Exploration  of  the  Electromagnetic  Field. 

(47)  Let  us  now  suppose  the  primary  circuit  A  to  be  of  invariable  form,  and  let  us 
explore  the  electromagnetic  field  by  means  of  the  secondary  circuit  B,  which  we  shall 
suppose  to  be  variable  in  form  and  position. 

We  may  begin  by  supposing  B  to  consist  of  a  short  straight  conductor  with  its  extre- 
mities sliding  on  two  parallel  conducting  rails,  which  are  put  in  connexion  at  some 
distance  from  the  sliding-piece. 



Then,  if  sliding  the  moveable  conductor  in  a  given  direction  increases  the  value  of  M, 
a  negative  electromotive  force  will  act  in  the  circuit  B,  tending  to  produce  a  negative 
current  in  B  during  the  motion  of  the  sliding-piece. 

If  a  current  be  kept  up  in  the  circuit  B,  then  the  sliding-piece  will  itself  tend  to 
move  in  that  direction,  which  causes  M  to  increase.  At  every  point  of  the  field  there 
will  always  be  a  certain  direction  such  that  a  conductor  moved  in  that  direction  does 
not  experience  any  electromotive  force  in  whatever  direction  its  extremities  are  turned. 
A  conductor  carrying  a  current  will  experience  no  mechanical  force  urging  it  in  that 
direction  or  the  opposite. 

This  direction  is  called  the  direction  of  the  line  of  magnetic  force  through  that  point. 

Motion  of  a  conductor  across  such  a  line  produces  electromotive  force  in  a  direction 
perpendicular  to  the  line  and  to  the  direction  of  motion,  and  a  conductor  carrying  a 
current  is  urged  in  a  direction  perpendicular  to  the  line  and  to  the  direction  of  the 

(48)  We  may  next  suppose  B  to  consist  of  a  very  small  plane  circuit  capable  of  being 
placed  in  any  position  and  of  having  its  plane  turned  in  any  direction.  The  value  of  M 
will  be  greatest  when  the  plane  of  the  circuit  is  perpendicular  to  the  line  of  magnetic 
force.  Hence  if  a  current  is  maintained  in  B  it  will  tend  to  set  itself  in  this  position, 
and  will  of  itself  indicate,  like  a  magnet,  the  direction  of  the  magnetic  force. 

On  Lines  of  Magnetic  Force. 

(49)  Let  any  surface  be  drawn,  cutting  the  lines  of  magnetic  force,  and  on  this  sur- 
face let  any  system  of  lines  be  drawn  at  small  intervals,  so  as  to  lie  side  by  side  without 
cutting  each  other.  Next,  let  any  line  be  drawn  on  the  surface  cutting  all  these  lines, 
and  let  a  second  line  be  drawn  near  it,  its  distance  from  the  first  being  such  that  the 
value  of  M  for  each  of  the  small  spaces  enclosed  between  these  two  lines  and  the  lines 
of  the  first  system  is  equal  to  unity. 

In  this  way  let  more  lines  be  drawn  so  as  to  form  a  second  system,  so  that  the  value  of 
M  for  every  reticulation  formed  by  the  intersection  of  the  two  systems  of  lines  is  unity. 

Finally,  from  every  point  of  intersection  of  these  reticulations  let  a  line  be  drawn 
through  the  field,  always  coinciding  in  direction  with  the  direction  of  magnetic  force. 

(50)  In  this  way  the  whole  field  will  be  filled  with  lines  of  magnetic  force  at  regular 
intervals,  and  the  properties  of  the  electromagnetic  field  will  be  completely  expressed 
by  them. 

For,  1st,  If  any  closed  curve  be  drawn  in  the  field,  the  value  of  M  for  that  curve  will 
be  expressed  by  the  number  of  lines  of  force  which  pass  through  that  closed  curve. 

2ndly.  If  this  curve  be  a  conducting  circuit  and  be  moved  through  the  field,  an 
electromotive  force  will  act  in  it,  represented  by  the  rate  of  decrease  of  the  number  of 
lines  passing  through  the  curve. 

3rdly.  If  a  current  be  maintained  in  the  circuit,  the  conductor  will  be  acted  on  by 
forces  tending  to  move  it  so  as  to  increase  the  number  of  lines  passing  through  it,  and 


the  amount  of  work  done  by  these  forces  is  equal  to  the  current  in  the  circuit  multi- 
plied by  the  number  of  additional  lines. 

4thly.  If  a  small  plane  circuit  be  placed  in  the  field,  and  be  free  to  turn,  it  will  place 
its  plane  perpendicular  to  the  lines  of  force.  A  small  magnet  will  place  itself  with  its 
axis  in  the  direction  of  the  lines  of  force. 

othly.  If  a  long  uniformly  magnetized  bar  is  placed  in  the  field,  each  pole  will  be 
acted  on  by  a  force  in  the  direction  of  the  lines  of  force.  The  number  of  lines  of  force 
passing  through  unit  of  area  is  equal  to  the  force  acting  on  a  unit  pole  multiplied  by  a 
coefficient  depending  on  the  magnetic  nature  of  the  medium,  and  called  the  coefficient 
of  magnetic  induction. 

In  fluids  and  isotropic  solids  the  value  of  this  coefficient  p  is  the  same  in  whatever 
direction  the  lines  of  force  pass  through  the  substance,  but  in  crystallized,  strained,  and 
organized  solids  the  value  of  p  may  depend  on  the  direction  of  the  lines  of  force  with 
respect  to  the  axes  of  crystallization,  strain,  or  growth. 

In  all  bodies  \l  is  affected  by  temperature,  and  in  iron  it  appears  to  diminish  as  the 
intensity  of  the  magnetization  increases. 

On  Magnetic  Equipotential  Surfaces. 

(51)  If  we  explore  the  field  with  a  uniformly  magnetized  bar,  so  long  that  one  of  its 
poles  is  in  a  very  weak  part  of  the  magnetic  field,  then  the  magnetic  forces  will  perform 
work  on  the  other  pole  as  it  moves  about  the  field. 

If  we  start  from  a  given  point,  and  move  this  pole  from  it  to  any  other  point,  the 
work  performed  will  be  independent  of  the  path  of  the  pole  between  the  two  points ; 
provided  that  no  electric  current  passes  between  the  different  paths  pursued  by  the  pole. 

Hence,  when  there  are  no  electric  currents  but  only  magnets  in  the  field,  we  may 
draw  a  series  of  surfaces  such  that  the  work  done  in  passing  from  one  to  another  shall 
be  constant  whatever  be  the  path  pursued  between  them.  Such  surfaces  are  called 
Equipotential  Surfaces,  and  in  ordinary  cases  are  perpendicular  to  the  Lines  of  mag- 
netic force. 

If  these  surfaces  are  so  drawn  that,  when  a  unit  pole  passes  from  any  one  to  the 
next  in  order,  unity  of  work  is  done,  then  the  work  done  in  any  motion  of  a  magnetic 
pole  will  be  measured  by  the  strength  of  the  pole  multiplied  by  the  number  of  surfaces 
which  it  has  passed  through  in  the  positive  direction. 

(52)  If  there  are  circuits  carrying  electric  currents  in  the  field,  then  there  will  still 
be  equipotential  surfaces  in  the  parts  of  the  field  external  to  the  conductors  carrying  the 
currents,  but  the  work  done  on  a  unit  pole  in  passing  from  one  to  another  will  depend 
on  the  number  of  times  which  the  path  of  the  pole  circulates  round  any  of  these 
currents.  Hence  the  potential  in  each  surface  will  have  a  series  of  values  in  arith- 
metical progression,  differing  by  the  work  done  in  passing  completely  round  one  of  the 
currents  in  the  field. 

The  equipotential  surfaces  will  not  be  continuous  closed  surfaces,  but  some  of  them 


will  be  limited  sheets,  terminating  in  the  electric  circuit  as  their  common  edge  or 
boundary.  The  number  of  these  will  be  equal  to  the  amount  of  work  done  on  a  unit 
pole  in  going  round  the  current,  and  this  by  the  ordinary  measurement  =4^ry,  where  y 
is  the  value  of  the  current. 

These  surfaces,  therefore,  are  connected  with  the  electric  current  as  soap-bubbles  are 
connected  with  a  ring  in  M.  Plateau's  experiments.  Every  current  y  has  4-z-y  surfaces 
attached  to  it.  These  surfaces  have  the  current  for  their  common  edge,  and  meet  it  at 
equal  angles.  The  form  of  the  surfaces  in  other  parts  depends  on  the  presence  of  other 
currents  and  magnets,  as  well  as  on  the  shape  of  the  circuit  to  which  they  belong. 


(53.)  Let  us  assume  three  rectangular  directions  in  space  as  the  axes  of  x,  y,  and  z, 

and  let  all  quantities  having  direction  be  expressed  by  their  components  in  these  three 


Electrical  Currents  (p,  q,  r). 

(54)  An  electrical  current  consists  in  the  transmission  of  electricity  from  one  part  of 
a  body  to  another.  Let  the  quantity  of  electricity  transmitted  in  unit  of  time  across 
unit  of  area  perpendicular  to  the  axis  of  x  be  called  p,  then  p  is  the  component  of  the 
current  at  that  place  in  the  direction  of  x. 

We  shall  use  the  letters  p,  q,  r  to  denote  the  components  of  the  current  per  unit  of 
area  in  the  directions  of  x,  y,  z. 

Electrical  Displacements  (f,  g,  h). 

(55)  Electrical  displacement  consists  in  the  opposite  electrification  of  the  sides  of  a 
molecule  or  particle  of  a  body  which  may  or  may  not  be  accompanied  with  transmission 
through  the  body.  Let  the  quantity  of  electricity  which  would  appear  on  the  faces  of  an  element  dx,  dy,  dz  cut  from  the  body  be,  then  f  is  the  component 
of  electric  displacement  parallel  to  x.  We  shall  use  f,  g,  h  to  denote  the  electric 
displacements  parallel  to  x,  y,  z  respectively. 

The  variations  of  the  electrical  displacement  must  be  added  to  the  currents  p,  q,  r  to 
get  the  total  motion  of  electricity,  which  we  may  call  p',  q',  r1,  so  that 

*=!  +  % 


Electromotive  Force  (P,  Q,  R). 
(56)  Let  P,  Q,  R  represent  the  components  of  the  electromotive  force  at  any  point. 
Then   P  represents  the  difference  of  potential  per  unit  of  length  in  a  conductor 


placed  in  the  direction  of  x  at  the  given  point.  We  may  suppose  an  indefinitely  short 
wire  placed  parallel  to  x  at  a  given  point  and  touched,  during  the  action  of  the  force  P, 
by  two  small  conductors,  which  are  then  insulated  and  removed  from  the  influence  of 
the  electromotive  force.  The  value  of  P  might  then  be  ascertained  by  measuring  the 
charge  of  the  conductors. 

Thus  if  I  be  the  length  of  the  wire,  the  difference  of  potential  at  its  ends  will  be  PZ, 
and  if  C  be  the  capacity  of  each  of  the  small  conductors  the  charge  on  each  will  be 
\CYl.  Since  the  capacities  of  moderately  large  conductors,  measured  on  the  electro- 
magnetic system,  are  exceedingly  small,  ordinary  electromotive  forces  arising  from 
electromagnetic  actions  could  hardly  be  measured  in  this  way.  In  practice  such  measure- 
ments are  always  made  with  long  conductors,  forming  closed  or  nearly  closed  circuits. 

Electromagnetic  Momentum  (F,  G,  H). 

(57)  Let  F,  G,  II  represent  the  components  of  electromagnetic  momentum  at  any 
point  of  the  field,  due  to  any  system  of  magnets  or  currents. 

Then  F  is  the  total  impulse  of  the  electromotive  force  in  the  direction  of  x  that  would 
be  generated  by  the  removal  of  these  magnets  or  currents  from  the  field,  that  is,  if  P 
be  the  electromotive  force  at  any  instant  during  the  removal  of  the  system 


Hence  the  part  of  the  electromotive  force  which  depends  on  the  motion  of  magnets  or 
currents  in  the  field,  or  their  alteration  of  intensity,  is 

P=_**,     Q=-^,    R=-S (29) 

dt  dt  .     dt 

Electromagnetic  Momentum  of  a  Circuit. 

(58)  Let  s  be  the  length  of  the  circuit,  then  if  we  integrate 

J(F|+G|+Il|j* (30) 

round  the  circuit,  we  shall  get  the  total  electromagnetic  momentum  of  the  circuit,  or  the 
number  of  lines  of  magnetic  force  which  pass  through  it,  the  variations  of  which  measure 
the  total  electromotive  force  in  the  circuit.  This  electromagnetic  momentum  is  the 
same  thing  to  which  Professor  Faeaday  has  applied  the  name  of  the  Electrotonic  State. 
If  the  circuit  be  the  boundary  of  the  elementary  area  dy  dz,  then  its  electromagnetic 
momentum  is 

and  this  is  the  number  of  lines  of  magnetic  force  which  pass  through  the  area  dy  dz. 

Magnetic  Force  (a,  /3,  y). 

(59)  Let  a,  /3,  y  represent  the  force  acting  on  a  unit  magnetic  pole  placed  at  the 
given  point  resolved  in  the  directions  of  x,  y,  and  z. 



Coefficient  of  Magnetic  Induction  (p). 

(60)  Let  ^  be  the  ratio  of  the  magnetic  induction  in  a  given  medium  to  that  in  air 
under  an  equal  magnetizing  force,  then  the  number  of  lines  of  force  in  unit  of  area 
perpendicular  to  x  will  be  pa  (p  is  a  quantity  depending  on  the  nature  of  the  medium, 
its  temperature,  the  amount  of  magnetization  already  produced,  and  in  crystalline  bodies 
varying  with  the  direction). 

(61)  Expressing  the  electric  momentum  of  small  circuits  perpendicular  to  the  three 
axes  in  this  notation,  we  obtain  the  following 

Equations  of  Magnetic  Force. 







dy  ' 


Equations  of  Currents. 

(62)  It  is  known  from  experiment  that  the  motion  of  a  magnetic  pole  in  the  electro- 
magnetic field  in  a  closed  circuit  cannot  generate  work  unless  the  circuit  which  the  pole 
describes  passes  round  an  electric  current.     Hence,  except  in  the  space  occupied  by  the 

electric  currents, 

adx-\-fidy-\-ydz=d<p (31) 

a  complete  differential  of  <p,  the  magnetic  potential. 

The  quantity  <p  may  be  susceptible  of  an  indefinite  number  of  distinct  values,  according 
to  the  number  of  times  that  the  exploring  point  passes  round  electric  currents  in  its 
course,  the  difference  between  successive  values  of  <p  corresponding  to  a  passage  com- 
pletely round  a  current  of  strength  c  being  4src. 

Hence  if  there  is  no  electric  current, 

dy       dz 

d3-  —  — =4t  ' 
dy       dz  " " 

dx       dy .      , 

dz       dx  * ' 

dfi      dx_        . 
dx      dy 

We  may  call  these  the  Equations  of  Currents. 

but  if  there  is  a  current  p', 



Electromotive  Force  in  a  Circuit. 
(63)  Let  |  be  the  electromotive  force  acting  round  the  circuit  A,  then 

5=.f(4t+Qt+4>> (32) 

where  ds  is  the  element  of  length,  and  the  integration  is  performed  round  the  circuit. 

Let  the  forces  in  the  field  be  those  due  to  the  circuits  A  and  B,  then  the  electro- 
magnetic momentum  of  A  is 

J(F§+Gt+Hs)*=L«+M»' (33) 

where  u  and  v  are  the  currents  in  A  and  B,  and 

s=-jt(Lu+m) (34) 

Hence,  if  there  is  no  motion  of  the  circuit  A, 

~       dt        dx 

U—  ~~  dt  ~dyy 


dt       dz 


where  "¥"  is  a  function  of  x,  y,  z,  and  t,  which  is  indeterminate  as  far  as  regards  the 
solution  of  the  above  equations,  because  the  terms  depending  on  it  will  disappear  on 
integrating  round  the  circuit.  The  quantity  "¥"  can  always,  however,  be  determined  in 
any  particular  case  when  we  know  the  actual  conditions  of  the  question.  The  physical 
interpretation  of  "¥"  is,  that  it  represents  the  electric  potential  at  each  point  of  space. 

Electromotive  Force  on  a  Moving  Conductor. 
(64)  Let  a  short  straight  conductor  of  length  a,  parallel  to  the  axis  of  w,  move  with 
a  velocity  whose  components  are  -^,  -J-,  -- ',  and  let  its  extremities  slide  along  two 

parallel  conductors  with  a  velocity  — .  Let  us  find  the  alteration  of  the  electro- 
magnetic momentum  of  the  circuit  of  which  this  arrangement  forms  a  part. 

In  unit  of  time  the  moving  conductor  has  travelled  distances  _ ,  Jf    £z  along:  the 

dt     at     dt  ° 

directions  of  the  three  axes,  and  at  the  same  time  the  lengths  of  the  parallel  conductors 

included  in  the  circuit  have  each  been  increased  by  -r.- 

Hence  the  quantity 

J  \     as  as  as/ 



will  be  increased  by  the  following  increments, 

/dF  doc  ,  dF  dy  .  dF  dz\     ,        ,  ,  •  r  , 

a    _--_-+-__  -±.-4-  —  _      due  to  motion  of  conductor, 
\dx  dt      dy  dt       dz  dtJ 

ds /dF  dx  .  dG  dy  ,  dH.  dz\     -,        ,     ■,        ,,       .  c    . 

-aAdx  Ts+-E  Ts+Tx  dsh  duG  t0  le^^enmg  of  circmt. 

The  total  increment  will  therefore  be 

/dF_dG\di_     /dll     dF\d~_ 
a  \dy      dx) dt     a\dx      dzjdt  ' 

or,  by  the  equations  of  Magnetic  Force  (8), 


If  P  is  the  electromotive  force  in  the  moving  conductor  parallel  to  x  referred  to  unit 
of  length,  then  the  actual  electromotive  force  is  Va ;  and  since  this  is  measured  by  the 
decrement  of  the  electromagnetic  momentum  of  the  circuit,  the  electromotive  force  due 
to  motion  will  be 

P=w|-^| («•) 

(65)  The  complete  equations  of  electromotive  force  on  a  moving  conductor  may  now 
be  written  as  follows : — 

Equations  of  Electromotive  Force. 

_  /    dy      ndz\       dF      d^ 


~         (    dz         dx\      dG      d<¥ 


-r,  I '  ndx         dy\      d 



dx      _  dy\      dK_dVi 
it  ~  dz 

The  first  term  on  the  right-hand  side  of  each  equation  represents  the  electromotive 
force  arising  from  the  motion  of  the  conductor  itself.  This  electromotive  force  is  per- 
pendicular to  the  direction  of  motion  and  to  the  lines  of  magnetic  force ;  and  if  a 
parallelogram  be  drawn  whose  sides  represent  in  direction  and  magnitude  the  velocity 
of  the  conductor  and  the  magnetic  induction  at  that  point  of  the  field,  then  the  area  of 
the  parallelogram  will  represent  the  electromotive  force  due  to  the  motion  of  the  con- 
ductor, and  the  direction  of  the  force  is  perpendicular  to  the  plane  of  the  parallelogram. 

The  second  term  in  each  equation  indicates  the  effect  of  changes  in  the  position  or 
strength  of  magnets  or  currents  in  the  field. 

The  third  term  shows  the  effect  of  the  electric  potential  Y.  It  has  no  effect  in 
causing  a  circulating  current  in  a  closed  circuit.  It  indicates  the  existence  of  a  force 
urging  the  electricity  to  or  from  certain  definite  points  in  the  field. 


Electric  Elasticity. 

(66)  When  an  electromotive  force  acts  on  a  dielectric,  it  puts  every  part  of  the 
dielectric  into  a  polarized  condition,  in  which  its  opposite  sides  are  oppositely  electri- 
fied. The  amount  of  this  electrification  depends  on  the  electromotive  force  and  on  the 
nature  of  the  substance,  and,  in  solids  having  a  structure  defined  by  axes,  on  the  direc- 
tion of  the  electromotive  force  with  respect  to  these  axes.  In  isotropic  substances,  if  k 
is  the  ratio  of  the  electromotive  force  to  the  electric  displacement,  we  may  write  the 

Equations  of  Electric  Elasticity, 

Q.=%,  [■  (E) 

Electric  Resistance. 

(67)  When  an  electromotive  force  acts  on  a  conductor  it  produces  a  current  of  elec- 
tricity through  it.  This  effect  is  additional  to  the  electric  displacement  already  con- 
sidered. In  solids  of  complex  structure,  the  relation  between  the  electromotive  force 
and  the  current  depends  on  their  direction  through  the  solid.  In  isotropic  substances, 
which  alone  we  shall  here  consider,  if  g>  is  the  specific  resistance  referred  to  unit  of 
volume,  we  may  write  the 

Equations  of  Electric  Resistance, 


Electric  Quantity. 

(68)  Let  e  represent  the  quantity  of  free  positive  electricity  contained  in  unit  of 
volume  at  any  part  of  the  field,  then,  since  this  arises  from  the  electrification  of  the 
different  parts  of  the  field  not  neutralizing  each  other,  we  may  write  the 

Equation  of  Free  Electricity, 

«+!+!+§=» (G) 

(69)  If  the  medium  conducts  electricity,  then  we  shall  have  another  condition,  which 
may  be  called,  as  in  hydrodynamics,  the 

Equation  of  Continuity, 
de,dp,dqdr_()  (m 

(70)  In  these  equations  of  the  electromagnetic  field  we  have  assumed  twenty  variable 
















quantities,  namely, 

For  Electromagnetic  Momentum F 

Magnetic  Intensity        a 

Electromotive  Force P 

Current  due  to  true  conduction j) 

Electric  Displacement f 

Total  Current  (including  variation  of  displacement)    .     .  p' 

Quantity  of  free  Electricity e 

Electric  Potential ~*¥ 

Between  these  twenty  quantities  we  have  found  twenty  equations,  viz. 

Three  equations  of  Magnetic  Force (B) 

Electric  Currents (C) 

Electromotive  Force (D) 

Electric  Elasticity (E) 

Electric  Resistance (F) 

Total  Currents (A) 

One  equation  of  Free  Electricity (G) 

„  Continuity (H) 

These  equations  are  therefore  sufficient  to  determine  all  the  quantities  which  occur 
in  them,  provided  we  know  the  conditions  of  the  problem.  In  many  questions,  how- 
ever, only  a  few  of  the  equations  are  required. 

Intrinsic  Energy  of  the  Electromagnetic  Field. 

(71)  We  have  seen  (33)  that  the  intrinsic  energy  of  any  system  of  currents  is  found 
by  multiplying  half  the  current  in  each  circuit  into  its  electromagnetic  momentum. 
This  is  equivalent  to  finding  the  integral 

E=i2(Fy+G£'+Hr')dV (37) 

over  all  the  space  occupied  by  currents,  where  jp,  q,  r  are  the  components  of  currents, 
and  F,  G,  H  the  components  of  electromagnetic  momentum. 

Substituting  the  values  of  p',  q',  r'  from  the  equations  of  Currents  (C),  this  becomes 


Integrating  by  parts,  and  remembering  that  a,  /3,  y  vanish  at  an  infinite  distance,  the 
expression  becomes 

Referring  to  the  equations  of 

1  -  f   /rfH     dG\  ,  _/rfF     dH\  .     fdG 

where  the  integration  is  to  be  extended  over  all  space. 
Magnetic  Force  (B),  p.  482,  this  becomes 




where  a,  /3,  7  are  the  components  of  magnetic  intensity  or  the  force  on  a  unit  magnetic 
pole,  and  ^a,  (jbfi,  [/y  are  the  components  of  the  quantity  of  magnetic  induction,  or  the 
number  of  lines  of  force  in  unit  of  area. 

In  isotropic  media  the  value  of  \x,  is  the  same  in  all  directions,  and  we  may  express 
the  result  more  simply  by  saying  that  the  intrinsic  energy  of  any  part  of  the  magnetic 
field  arising  from  its  magnetization  is 

i4    T2 

per  unit  of  volume,  where  I  is  the  magnetic  intensity. 

(72)  Energy  may  be  stored  up  in  the  field  in  a  different  way,  namely,  by  the  action 
of  electromotive  force  in  producing  electric  displacement.  The  work  done  by  a  variable 
electromotive  force,  P,  in  producing  a  variable  displacement,  f,  is  got  by  integrating 

from  P  =  0  to  the  given  value  of  P. 

Since  ~P=kf,  equation  (E),  this  quantity  becomes 

Hence  the  intrinsic  energy  of  any  part  of  the  field,  as  existing  in  the  form  of  electric 
displacement,  is 

The  total  energy  existing  in  the  field  is  therefore 

E=2{^(a^+^/3+y^y)+i(P/,+Qy+EA)}^V (I) 

The  first  term  of  this  expression  depends  on  the  magnetization  of  the  field,  and  is 
explained  on  our  theory  by  actual  motion  of  some  kind.  The  second  term  depends  on 
the  electric  polarization  of  the  field,  and  is  explained  on  our  theory  by  strain  of  some 
kind  in  an  elastic  medium. 

(73)  I  have  on  a  former  occasion  *  attempted  to  describe  a  particular  kind  of  motion 
and  a  particular  kind  of  strain,  so  arranged  as  to  account  for  the  phenomena.  In  the 
present  paper  I  avoid  any  hypothesis  of  this  kind ;  and  in  using  such  words  as  electric 
momentum  and  electric  elasticity  in  reference  to  the  known  phenomena  of  the  induc- 
tion of  currents  and  the  polarization  of  dielectrics,  I  wish  merely  to  direct  the  mind  of 
the  reader  to  mechanical  phenomena  which  will  assist  him  in  understanding  the  elec- 
trical ones.  All  such  phrases  in  the  present  paper  are  to  be  considered  as  illustrative, 
not  as  explanatory. 

(71)  In  speaking  of  the  Energy  of  the  field,  however,  I  wish  to  be  understood  literally. 
All  energy  is  the  same  as  mechanical  energy,  whether  it  exists  in  the  form  of  motion  or 
in  that  of  elasticity,  or  in  any  other  form.  The  energy  in  electromagnetic  phenomena  is 
mechanical  energy.     The  only  question  is,  Where  does  it  reside  \     On  the  old  theories 

*  "On  Physical  Lines  of  Eorce,"  Philosophical  Magazine,  1861-62. 


it  resides  in  the  electrified  bodies,  conducting  circuits,  and  magnets,  in  the  form  of  an 
unknown  quality  called  potential  energy,  or  the  power  of  producing  certain  effects  at  a 
distance.  On  our  theory  it  resides  in  the  electromagnetic  field,  in  the  space  surrounding 
the  electrified  and  magnetic  bodies,  as  well  as  in  those  bodies  themselves,  and  is  in  two 
different  forms,  which  may  be  described  without  hypothesis  as  magnetic  polarization 
and  electric  polarization,  or,  according  to  a  very  probable  hypothesis,  as  the  motion  and 
the  strain  of  one  and  the  same  medium. 

(75)  The  conclusions  arrived  at  in  the  present  paper  are  independent  of  this  hypo- 
thesis, being  deduced  from  experimental  facts  of  three  kinds : — 

1.  The  induction  of  electric  currents  by  the  increase  or  diminution  of  neighbouring 
currents  according  to  the  changes  in  the  lines  of  force  passing  through  the  circuit. 

2.  The  distribution  of  magnetic  intensity  according  to  the  variations  of  a  magnetic 

3.  The  induction  (or  influence)  of  statical  electricity  through  dielectrics. 

We  may  now  proceed  to  demonstrate  from  these  principles  the  existence  and  laws  of 
the  mechanical  forces  which  act  upon  electric  currents,  magnets,  and  electrified  bodies 
placed  in  the  electromagnetic  field. 

Mechanical  Force  on  a  Moveable  Conductor. 

(76)  We  have  shown  (§§  34  &  35)  that  the  work  done  by  the  electromagnetic  forces 
in  aiding  the  motion  of  a  conductor  is  equal  to  the  product  of  the  current  in  the  con- 
ductor multiplied  by  the  increment  of  the  electromagnetic  momentum  due  to  the 

Let  a  short  straight  conductor  of  length  a  move  parallel  to  itself  in  the  direction  of 
X,  with  its  extremities  on  two  parallel  conductors.  Then  the  increment  of  the  electro- 
magnetic momentum  due  to  the  motion  of  a  will  be 

/  JF  dx       r/G  dy      dK  dz\^ 
'  I  dx    ds    '    dx   ds   '    dx   ds  J 

That  due  to  the  lengthening  of  the  circuit  by  increasing  the  length  of  the  parallel  con- 
ductors will  be 

The  total  increment  is 

/d¥  dx      dFdy    ,^F^\v 
~a\dx  ^+  dy  ds~T~  dz    ds)bX- 

a8x\ds  \dx  ~  dy  )  ~  ds\dz        dx  ) 

which  is  by  the  equations  of  Magnetic  Force  (B),  p.  482, 


(|w- 10 

Let  X  be  the  force  acting  along  the  direction  of  x  per  unit  of  length  of  the  conductor, 
then  the  work  done  is  X«k 


Let  C  be  the  current  in  the  conductor,  and  letj/,  q',  r'  be  its  components,  then 

or  X.=[vyql  — ffcj3/. ' 

Similarly,  Y=[Aotr'  —  ^yp\ }•       •     ■     •     •  ••;   •, .■ (J) 

These  are  the  equations  which  determine  the  mechanical  force  acting  on  a  conductor 
carrying  a  current.  The  force  is  perpendicular  to  the  current  and  to  the  lines  of  force, 
and  is  measured  by  the  area  of  the  parallelogram  formed  by  lines  parallel  to  the  current 
and  lines  of  force,  and  proportional  to  their  intensities. 

Mechanical  Force  on  a  Magnet. 

(77)  In  any  part  of  the  field  not  traversed  by  electric  currents  the  distribution  of 

magnetic  intensity  may  be  represented  by  the   differential   coefficients  of  a  function 

which  may  be  called  the  magnetic  potential.     When  there  are  no  currents  in  the  field, 

this  quantity  has  a  single  value  for  each  point.     When  there  are  currents,  the  potential 

has  a  series  of  values  at  each  point,  but  its  differential  coefficients  have  only  one  value, 


dip  d<p  dip 

fa=a>       ~dy=^      Tz=V- 

Substituting  these  values  of  a,  p,  y  in  the  expression  (equation  38)  for  the  intrinsic 
energy  of  the  field,  and  integrating  by  parts,  it  becomes 

The  expression 

*(£+£ +&).**=*** m 

indicates  the  number  of  lines  of  magnetic  force  which  have  their  origin  within  the 
space  V.  Now  a  magnetic  pole  is  known  to  us  only  as  the  origin  or  termination  of 
lines  of  magnetic  force,  and  a  unit  pole  is  one  which  has  4r  lines  belonging  to  it,  since 
it  produces  unit  of  magnetic  intensity  at  unit  of  distance  over  a  sphere  whose  surface 
is  4t. 

Hence  if  m  is  the  amount  of  free  positive  magnetism  in  unit  of  volume,  the  above 
expression  may  be  written  4&m,  and  the  expression  for  the  energy  of  the  field  becomes 

■E=-%(±$m)dV (40) 

If  there  are  two  magnetic  poles  m,  .and  m2  producing  potentials  p,  and  <p2  in  the  field  , 
then  if  m2  is  moved  a  distance  dx,  and  is  urged  in  that  direction  by  a  force  X,  then  the 
work  done  is  ILdx,  and  the  decrease  of  energy  in  the  field  is 

<7(iOi +  £2X^.4-^)), 
and  these  must  be  equal  by  the  principle  of  Conservation  of  Energy. 


Since  the  distribution  <p,  is  determined  by  m15  and  <p2  by  m2,  the  quantities  ftm,  and 
<p2  ra2  will  remain  constant. 

It  can  be  shown  also,  as  Green  has  proved  (Essay,  p.  10),  that 

so  that  we  get 




where  a,  represents  the  magnetic  intensity  due  to  mv 
Similarly,  Y=m2j3i, 



So  that  a  magnetic  pole  is  urged  in  the  direction  of  the  lines  of  magnetic  force  with 
a  force  equal  to  the  product  of  the  strength  of  the  pole  and  the  magnetic  intensity. 

(78)  If  a  single  magnetic  pole,  that  is  one  pole  of  a  very  long  magnet,  be  placed  in 
the  field,  the  only  solution  of  <p  is 

ft—?? («) 

r  r 
where  m,  is  the  strength  of  the  pole  and  r  the  distance  from  it. 
The  repulsion  between  two  poles  of  strength  m1  and  m2  is 

dp^m^ (42) 

dr         i*-rz  v      ' 

In  air  or  any  medium  in  which  |0,=1  this  is  simply  — L-?>  but  in  other  media  the  force 

acting  between  two  given  magnetic  poles  is  inversely  proportional  to  the  coefficient  of 
magnetic  induction  for  the  medium.  This  may  be  explained  by  the  magnetization  of 
the  medium  induced  by  the  action  of  the  poles. 

Mechanical  Force  on  an  Electrified  Body. 

(79)  If  there  is  no  motion  or  change  of  strength  of  currents  or  magnets  in  the  field, 
the  electromotive  force  is  entirely  due  to  variation  of  electric  potential,  and  we  shall 
have  (§65) 

P=  —  — ,    Q=r— '— »    R=  —  — 

dx  dij  dz 

Integrating  by  parts  the  expression  (I)  for  the  energy  due  to  electric  displacement,  and 
remembering  that  P,  Q,  E,  vanish  at  an  infinite  distance,  it  becomes 


or  by  the  equation  of  Free  Electricity  (G),  p.  485, 



By  the  same  demonstration  as  was  used  in  the  case  of  the  mechanical  action  on  a  magnet, 
it  may  be  shown  that  the  mechanical  force  on  a  small  body  containing  a  quantity  e2  of 
free  electricity  placed  in  a  field  whose  potential  arising  from  other  electrified  bodies 
is  y¥1,  has  for  components 

v-    dli-      n 

X~e2  dy  ~ ~ y^2 


So  that  an  electrified  body  is  urged  in  the  direction  of  the  electromotive  force  with  a 
force  equal  to  the  product  of  the  quantity  of  free  electricity  and  the  electromotive  force. 
If  the  electrification  of  the  field  arises  from  the  presence  of  a  small  electrified  body 
containing  e,  of  free  electrity,  the  only  solution  of  "*¥  x  is 

"¥1-±e-L, (43) 

where  r  is  the  distance  from  the  electrified  body. 

The  repulsion  between  two  electrified  bodies  e„  e2  is  therefore 

e  Wx  _  k  <v?2  ^ 

Measurement  of  Electrical  Phenomena  by  Electrostatic  Effects. 

(80)  The  quantities  with  which  we  have  had  to  do  have  been  hitherto  expressed  in 
terms  of  the  Electromagnetic  System  of  measurement,  which  is  founded  on  the  mecha- 
nical action  between  currents.  The  electrostatic  system  of  measurement  is  founded  on 
the  mechanical  action  between  electrified  bodies,  and  is  independent  of,  and  incom- 
patible with,  the  electromagnetic  system ;  so  that  the  units  of  the  different  kinds  of 
quantity  have  different  values  according  to  the  system  we  adopt,  and  to  pass  from  the 
one  system  to  the  other,  a  reduction  of  all  the  quantities  is  required. 

According  to  the  electrostatic  system,  the  repulsion  between  two  small  bodies  charged 
with  quantities  ^. ,  fj2  of  electricity  is 


where  r  is  the  distance  between  them. 

Let  the  relation  of  the  two  systems  be  such  that  one  electromagnetic  unit  of  elec- 
tricity contains  v  electrostatic  units ;  then  ^l=vel  and  tj2z=ve2,  and  this  repulsion  becomes 

«2^=^^by  equation  (44), (45) 

whence  k,  the  coefficient  of  "  electric  elasticity  "  in  the  medium  in  which  the  experi- 
ments are  made,  i.  e.  common  air,  is  related  to  v,  the  number  of  electrostatic  units  in  one 

electromagnetic  unit,  by  the  equation 

&=4*n;2 (46) 



The  quantity  v  may  be  determined  by  experiment  in  several  ways.  According  to  the 
experiments  of  MM.  Webek  and  Kohlrausch, 

v= 3 10,740,000  metres  per  second. 

(81)  It  appears  from  this  investigation,  that  if  we  assume  that  the  medium  which 
constitutes  the  electromagnetic  field  is,  when  dielectric,  capable  of  receiving  in  every 
part  of  it  an  electric  polarization,  in  which  the  opposite  sides  of  every  element  into 
which  we  may  conceive  the  medium  divided  are  oppositely  electrified,  and  if  we  also 
assume  that  this  polarization  or  electric  displacement  is  proportional  to  the  electro- 
motive force  which  produces  or  maintains  it,  then  we  can  show  that  electrified  bodies 
in  a  dielectric  medium  will  act  on  one  another  with  forces  obeying  the  same  laws  as  are 
established  by  experiment. 

The  energy,  by  the  expenditure  of  which  electrical  attractions  and  repulsions  are  pro- 
duced, we  suppose  to  be  stored  up  in  the  dielectric  medium  which  surrounds  the  electri- 
fied bodies,  and  not  on  the  surface  of  those  bodies  themselves,  which  on  our  theory 
are  merely  the  bounding  surfaces  of  the  air  or  other  dielectric  in  which  the  true  springs 
of  action  are  to  be  sought. 

Note  on  the  Attraction  of  Gravitation. 

(82)  After  tracing  to  the  action  of  the  surrounding  medium  both  the  magnetic  and 
the  electric  attractions  and  repulsions,  and  finding  them  to  depend  on  the  inverse  square 
of  the  distance,  we  are  naturally  led  to  inquire  whether  the  attraction  of  gravitation, 
which  follows  the  same  law  of  the  distance,  is  not  also  traceable  to  the  action  of  a 
surrounding  medium. 

Gravitation  differs  from  magnetism  and  electricity  in  this ;  that  the  bodies  concerned 
are  all  of  the  same  kind,  instead  of  being  of  opposite  signs,  like  magnetic  poles  and 
electrified  bodies,  and  that  the  force  between  these  bodies  is  an  attraction  and  not  a 
repulsion,  as  is  the  case  between  like  electric  and  magnetic  bodies. 

The  lines  of  gravitating  force  near  two  dense  bodies  are  exactly  of  the  same  form  as 
the  lines  of  magnetic  force  near  two  poles  of  the  same  name ;  but  whereas  the  poles  are 
repelled,  the  bodies  are  attracted.  Let  E  be  the  intrinsic  energy  of  the  field  surrounding 
two  gravitating  bodies  M,,  M2,  and  let  E'  be  the  intrinsic  energy  of  the  field  surrounding 
two  magnetic  poles  to,,  m2,  equal  in  numerical  value  to  M,,  M2,  and  let  X  be  the  gravi- 
tating force  acting  during  the  displacement  hx,  and  X'  the  magnetic  force, 

X^=SE,        X'fcF=&E'; 

now  X  and  X'  are  equal  in  numerical  value,  but  of  opposite  signs ;  so  that 





where  a,  0,  y  are  the  components  of  magnetic  intensity.  If  R  be  the  resultant  gravi- 
tating force,  and  R'  the  resultant  magnetic  force  at  a  corresponding  part  of  the  field, 

R=-R',  and  a2+/32+y2=R2=R'2. 

E=C-2^RW (47) 

The  intrinsic  energy  of  the  field  of  gravitation  must  therefore  be  less  wherever  there  is 
a  resultant  gravitating  force. 

As  energy  is  essentially  positive,  it  is  impossible  for  any  part  of  space  to  have  nega- 
tive intrinsic  energy.  Hence  those  parts  of  space  in  which  there  is  no  resultant  force, 
such  as  the  points  of  equilibrium  in  the  space  between  the  different  bodies  of  a  system, 
and  within  the  substance  of  each  body,  must  have  an  intrinsic  energy  per  unit  of  volume 

greater  than 

—  R2 

8*      ' 

where  R  is  the  greatest  possible  value  of  the  intensity  of  gravitating  force  in  any  part  of 
the  universe. 

The  assumption,  therefore,  that  gravitation  arises  from  the  action  of  the  surrounding 
medium  in  the  way  pointed  out,  leads  to  the  conclusion  that  every  part  of  this  medium 
possesses,  when  undisturbed,  an  enormous  intrinsic  energy,  and  that  the  presence  of 
dense  bodies  influences  the  medium  so  as  to  diminish  this  energy  wherever  there  is  a 
resultant  attraction. 

As  I  am  unable  to  understand  in  what  way  a  medium  can  possess  such  properties,  I 
cannot  go  any  further  in  this  direction  in  searching  for  the  cause  of  gravitation. 


Capacity  of  a  Condenser. 

(83)  The  simplest  form  of  condenser  consists  of  a  uniform  layer  of  insulating  matter 
bounded  by  two  conducting  surfaces,  and  its  capacity  is  measured  by  the  quantity  of 
electricity  on  either  surface  when  the  difference  of  potentials  is  unity. 

Let  S  be  the  area  of  either  surface,  a  the  thickness  of  the  dielectric,  and  Tc  its  coeffi- 
cient of  electric  elasticity ;  then  on  one  side  of  the  condenser  the  potential  is  M^,  and  on 
the  other  side  ^,+1,  and  within  its  substance 

£-:-# («) 

Since  -j-  and  therefore /is  zero  outside  the  condenser,  the  quantity  of  electricity  on  its 

first  surface  =  —  Sf  and  on  the  second  +S/.     The  capacity  of  the  condenser  is  there- 

fore  S/"=^  in  electromagnetic  measure. 



Specific  Capacity  of  Electric  Induction  (D). 

(84)  If  the  dielectric  of  the  condenser  be  air,  then  its  capacity  in  electrostatic  mea- 

sure  is  —  (neglecting  corrections  arising  from  the  conditions  to  be  fulfilled  at  the 

edges).     If  the  dielectric  have  a  capacity  whose  ratio  to  that  of  air  is  D,  then  the  capa- 


city  of  the  condenser  will  be  — • 


Hence  D=T» (49) 

where  k0  is  the  value  of  Jc  in  air,  which  is  taken  for  unity. 

Electric  Absorption. 

(85)  When  the  dielectric  of  which  the  condenser  is  formed  is  not  a  perfect  insulator, 
the  phenomena  of  conduction  are  combined  with  those  of  electric  displacement.  The 
condenser,  when  left  charged,  gradually  loses  its  charge,  and  in  some  cases,  after  being 
discharged  completely,  it  gradually  acquires  a  new  charge  of  the  same  sign  as  the  original 
charge,  and  this  finally  disappears.  These  phenomena  have  been  described  by  Professor 
Faraday  (Experimental  Researches,  Series  XI.)  and  by  Mr.  F.  Jenkin  (Report  of  Com- 
mittee of  Board  of  Trade  on  Submarine  Cables),  and  may  be  classed  under  the  name  of 
"Electric  Absorption." 

(86)  We  shall  take  the  case  of  a  condenser  composed  of  any  number  of  parallel  layers 
of  different  materials.  If  a  constant  difference  of  potentials  between  its  extreme 
surfaces  is  kept  up  for  a  sufficient  time  till  a  condition  of  permanent  steady  flow  of 
electricity  is  established,  then  each  bounding  surface  will  have  a  charge  of  electricity 
depending  on  the  nature  of  the  substances  on  each  side  of  it.  If  the  extreme  surfaces 
be  now  discharged,  these  internal  charges  will  gradually  be  dissipated,  and  a  certain 
charge  may  reappear  on  the  extreme  surfaces  if  they  are  insulated,  or,  if  they  are  con- 
nected by  a  conductor,  a  certain  quantity  of  electricity  may  be  urged  through  the  con- 
ductor during  the  reestablishment  of  equilibrium. 

Let  the  thickness  of  the  several  layers  of  the  condenser  be  a„  a2,  &c. 
Let  the  values  of  Jc  for;  these  layers  be  respectively  fc},  k2,  Jc3,  and  let 

a1k2+a2Jc2-{-Szc.=ak, (50) 

where  Jc  is  the  "  electric  elasticity"  of  air,  and  a  is  the  thickness  of  an  equivalent  con- 
denser of  air. 

Let  the  resistances  of  the  layers  be  respectively  r„  r2,  Sec,  and  let  rx-{-r2-\-  &c.  =r  be 
the  resistance  of  the  whole  condenser,  to  a  steady  current  through  it  per  unit  of  surface. 

Let  the  electric  displacement  in  each  layer  be  /„  f2,  Sec 

Let  the  electric  current  in  each  layer  hept,p2,  &c.  ; 

Let  the  potential  on  the  first  surface  be  M*,,  and  the  electricity  per  unit  of  surface  ey 

Let  the  corresponding  quantities  at  the  boundary  of  the  first  and  second  surface  be 
^a  and  e2,  and  so  on.     Then  by  equations  (G)  and  (H), 



„           dex 
•1— — Jii           <#  —     Pi-> 

ei  —J  1  "~/aj            "^"  — JP 1 "" "i,2» 

&c.                   &c. 

But  by  equations  (E)  and  (F), 


y2      ^3 = dJC^j^  =       Y^Pv) 


&C.            &C.                &C. 




After  the  electromotive  force  has  been  kept  up  for  a  sufficient  time  the  current 
becomes  the  same  in  each  layer,  and 

px=p.2=&c.  =p=  —, 

where  *&  is  the  total  difference  of  potentials  between  the  extreme  layers.    We  have  then 

......    (53) 


Jl  r  a.k, 

J  r  a2A:2 

6l~  r  a,*,' 

=  !(j±_Jl-\    &c 

2       r  \ajtz      akxJ'  , 

These  are  the  quantities  of  electricity  on  the  different  surfaces. 

(87)  Now  let  the  condenser  be  discharged  by  connecting  the  extreme  surfaces 
through  a  perfect  conductor  so  that  their  potentials  are  instantly  rendered  equal,  then 
the  electricity  on  the  extreme  surfaces  will  be  altered,  but  that  on  the  internal  surfaces 
will  not  have  time  to  escape.     The  total  difference  of  potentials  is  now 

V^ajc^+ajc^+e^+ajc^+es+ez),  &c.  =0, (54) 

whence  if  dx  is  what  ex  becomes  at  the  instant  of  discharge, 

1       r  «,£, 



ak      ei      ak' 


The  instantaneous  discharge  is  therefore  -j-,  or  the  quantity  which  would  be  dis- 
charged by  a  condenser  of  air  of  the  equivalent  thickness  a,  and  it  is  unaffected  by  the 
want  of  perfect  insulation. 

(88)  Now  let  us  suppose  the  connexion  between  the  extreme  surfaces  broken,  and  the 
condenser  left  to  itself,  and  let  us  consider  the  gradual  dissipation  of  the  internal  charges. 
Let  "¥'  be  the  difference  of  potential  of  the  extreme  surfaces  at  any  time  t ;  then 

Y'=a^,/i  +  «^+&c-' (56) 





Hence  fi=A1e~  r>    ,  f2=A2e~~^t ,  &c. ;  and  by  referring  to  the  values  of  e\,  e2,  &c., 
we  find 

1       r  flj^j      ak 


&c. ; 
so  that  we  find  for  the  difference  of  extreme  potentials  at  any  time, 

(89)  It  appears  from  this  result  that  if  all  the  layers  are  made  of  the  same  sub- 
stance, V  will  be  zero  always.  If  they  are  of  different  substances,  the  order  in  which 
they  are  placed  is  indifferent,  and  the  effect  will  be  the  same  whether  each  substance 
consists  of  one  layer,  or  is  divided  into  any  number  of  thin  layers  and  arranged  in  any 
order  among  thin  layers  of  the  other  substances.  Any  substance,  therefore,  the  parts 
of  which  are  not  mathematically  homogeneous,  though  they  may  be  apparently  so,  may 
exhibit  phenomena  of  absorption.  Also,  since  the  order  of  magnitude  of  the  coefficients 
is  the  same  as  that  of  the  indices,  the  value  of  "¥*  can  never  change  sign,  but  must  start 
from  zero,  become  positive,  and  finally  disappear. 

(90)  Let  us  next  consider  the  total  amount  of  electricity  which  would  pass  from  the 
first  surface  to  the  second,  if  the  condenser,  after  being  thoroughly  saturated  by  the 
current  and  then  discharged,  has  its  extreme  surfaces  connected  by  a  conductor  of 
resistance  R.     Let  p  be  the  current  in  this  conductor ;  then,  during  the  discharge, 

■¥'=^1r1+^2r-2+&c.=^R (59) 

Integrating  with  respect  to  the  time,  and  calling  qt,  q2i  q  the  quantities  of  electricity 
which  traverse  the  different  conductors, 

qfy+q^+Scc^qR (60) 

The  quantities  of  electricity  on  the  several  surfaces  will  be 

,  -•  4— 2— ft> 

i+ft— ft» 

&c. ; 

and  since  at  last  all  these  quantities  vanish,  we  find 

2}    ........    ft  =4+02— #; 

whence  ?R=f  (_L  +  _|-  +  &c.)-s, 


a  quantity  essentially  positive ;  so  that,  when  the  primary  electrification  is  in  one  direc- 
tion, the  secondary  discharge  is  always  in  the  same  direction  as  the  primary  discharge*. 


(91)  At  the  commencement  of  this  paper  we  made  use  of  the  optical  hypothesis  of 
an  elastic  medium  through  which  the  vibrations  of  light  are  propagated,  in  order  to 
show  that  we  have  warrantable  grounds  for  seeking,  in  the  same  medium,  the  cause  of 
other  phenomena  as  well  as  those  of  light.  We  then  examined  electromagnetic  pheno- 
mena, seeking  for  their  explanation  in  the  properties  of  the  field  which  surrounds  the 
electrified  or  magnetic  bodies.  In  this  way  we  arrived  at  certain  equations  expressing 
certain  properties  of  the  electromagnetic  field.  We  now  proceed  to  investigate  whether 
these  properties  of  that  which  constitutes  the  electromagnetic  field,  deduced  from  electro- 
magnetic phenomena  alone,  are  sufficient  to  explain  the  propagation  of  light  through 
the  same  substance. 

(92)  Let  us  suppose  that  a  plane  wave  whose  direction  cosines  are  I,  m,  n  is  propa- 
gated through  the  field  with  a  velocity  V.     Then  all  the  electromagnetic  functions  will 

be  functions  of  7    .         .  xr , 


The  equations  of  Magnetic  Force  (B),  p.  482,  will  become 

dU         dG 

~  aw         dw 

dV      .  dH 

u,B=n  -j 1  j — 5 

~  aw  aw 

dG  dF 

^'  dw  dw  .- 

If  we  multiply  these  equations  respectively  by  I,  m,  n,  and  add,  we  find 

/^«  +  m^(3+w//-y==0, (62) 

which  shows  that  the  direction  of  the  magnetization  must  be  in  the  plane  of  the  wave. 

(93)  If  we  combine  the  equations  of  Magnetic  Force  (B)  with  those  of  Electric 
Currents  (C),  and  put  for  brevity 

dF  .dG    m    T  d*     d*     d* 


4^/  =^-V2H. 


*  Since  this  paper  was  communicated  to  the  Royal  Society,  I  have  seen  a  paper  by  M.  Gaugadt  in  the  Annales 
de  Chimie  for  1864,  in  which  he  has  deduced  the  phenomena  of  electric  absorption  and  secondary  discharge 
from  the  theory  of  compound  condensers. 



If  the  medium  in  the  field  is  a  perfect  dielectric  there  is  no  true  conduction,  and  the 
currents  p\  q',  r*  are  only  variations  in  the  electric  displacement,  or,  by  the  equations  of 
Total  Currents  (A), 

i—df  n'—d9 

*  =2' 

-  ■  -  ,     dh 

q ~ft'  V~Jt' 


But  these  electric  displacements  are  caused  by  electromotive  forces,  and  by  the  equations 
of  Electric  Elasticity  (E), 

P=£/,  Q=kg,         B,=kh (66) 

These  electromotive  forces  are  due  to  the  variations  either  of  the  electromagnetic  or 
the  electrostatic  functions,  as  there  is  no  motion  of  conductors  in  the  field ;  so  that  the 
equations  of  electromotive  force  (D)  are 

dt        dx 

dQ     dW 

^~~dt       dy  ' 


dt        dz 


(94)  Combining  these  equations,  we  obtain  the  following :- 

•        *(S-vG)+^+g)=o, 



If  we  differentiate  the  third  of  these  equations  with  respect  to  y,  and  the  second  with 
respect  to  z,  and  subtract,  J  and  1£  disappear,  and  by  remembering  the  equations  (B)  of 
magnetic  force,  the  results  may  be  written 


d?     _ 


(95)  If  we  assume  that  a,  /3,  y  are  functions  of  lx-\-my-\-nz— Yt=w,  the  first  equa- 
tion becomes 

*£="**•£• (70) 


"~  — V    4*p, 


The  other  equations  give  the  same  value  for  V,  so  that  the  wave  is  propagated  in  either 
direction  with  a  velocity  V. 


This  wave  consists  entirely  of  magnetic  disturbances,  the  direction  of  magnetization 
being  in  the  plane  of  the  wave.  No  magnetic  disturbance  whose  direction  of  magneti- 
zation is  not  in  the  plane  of  the  wave  can  be  propagated  as  a  plane  wave  at  all. 

Hence  magnetic  disturbances  propagated  through  the  electromagnetic  field  agree  with 
light  in  this,  that  the  disturbance  at  any  point  is  transverse  to  the  direction  of  propaga- 
tion, and  such  waves  may  have  all  the  properties  of  polarized  light. 

(96)  The  only  medium  in  which  experiments  have  been  made  to  determine  the  value 
of  A'  is  air,  in  which  /a=l,  and  therefore,  by  equation  (46), 

V=v (72) 

By  the  electromagnetic  experiments  of  MM.  Weber  and  Kohlrausch*, 

v=  310,740,000  metres  per  second 

is  the  number  of  electrostatic  units  in  one  electromagnetic  unit  of  electricity,  and  this, 
according  to  our  result,  should  be  equal  to  the  velocity  of  light  in  air  or  vacuum. 
The  velocity  of  light  in  air,  by  M.  FrzEAu'sf  experiments,  is 


according  to  the  more  accurate  experiments  of  M.  Foucault  J, 


The  velocity  of  light  in  the  space  surrounding  the  earth,  deduced  from  the  coefficient 
of  aberration  and  the  received  value  of  the  radius  of  the  earth's  orbit,  is 

V=  308,000,000. 

(97)  Hence  the  velocity  of  light  deduced  from  experiment  agrees  sufficiently  well 
with  the  value  of  v  deduced  from  the  only  set  of  experiments  we  as  yet  possess.  The 
value  of  v  was  determined  by  measuring  the  electromotive  force  with  which  a  condenser 
of  known  capacity  was  charged,  and  then  discharging  the  condenser  through  a  galvano- 
meter, so  as  to  measure  the  quantity  of  electricity  in  it  in  electromagnetic  measure. 
The  only  use  made  of  light  in  the  experiment  was  to  see  the  instruments.  The  value 
of  V  found  by  M.  Foucault  was  obtained  by  determining  the  angle  through  which  a 
revolving  mirror  turned,  while  the  light  reflected  from  it  went  and  returned  along  a 
measured  course.     No  use  whatever  was  made  of  electricity  or  magnetism. 

The  agreement  of  the  results  seems  to  show  that  light  and  magnetism  are  affections 
of  the  same  substance,  and  that  light  is  an  electromagnetic  disturbance  propagated 
through  the  field  according  to  electromagnetic  laws. 

(98)  Let  us  now  go  back  upon  the  equations  in  (94),  in  which  the  quantities  J  and 
"f  occur,  to  see  whether  any  other  kind  of  disturbance  can  be  propagated  through 
the  medium  depending  on  these  quantities  which  disappeared  from  the  final  equations. 

*  Leipzig  Transactions,  vol.  v.  (1857),  p.  260,  or  Poggendokff's  '  Annalen,'  Aug.  1856,  p.  10. 
f  Comptes  Eendus,  vol.  xxix.  (1849),  p.  90.  J  Ibid.  vol.  lv.  (1862),  pp.  501,  792. 



If  we  determine  •&  from  the  equation 

and  F',  G',  H'  from  the  equations 

F'  =  F-^,     G'=G-^,     H'=H-^,     ....     (74) 

«.r  dy  dz  ^      ' 


and  the  equations  in  (94)  become  of  the  form 

***=**(£+».(*+$)) (76) 

Differentiating  the  three  equations  with  respect  to  x,  y,  and  z,  and  adding,  we  find  that 

*=-t +*(*>'*■*). (77) 

and  that  A; V2F'  =  4frp  ^ , 

*V2G'  =  4^^ (78) 


'     '"      j 


Hence  the  disturbances   indicated  by   F',    G',  H'  are  propagated  with  the  velocity 
V  =  \/ —  through  the  field;   and  since 

V      4TTfJL 

dx    '     dy    '     dx  ' 

the  resultant  of  these  disturbances  is  in  the  plane  of  the  wave. 

(99)  The  remaining  part  of  the  total  disturbances  F,  G,  H  being  the  part  depending 
on  %,  is  subject  to  no  condition  except  that  expressed  in  the  equation 

dt  "T"  dt*'—U' 
If  we  perform  the  operation  V2  on  this  equation,  it  becomes 

ke=§-kV^x,y,.z) (79) 

Since  the  medium  is  a  perfect  insulator,  e,  the  free  electricity,  is  immoveable,  and 

therefore  -j.  is  a  function  of  x,  y,  z,  and  the  value  of  J  is  either  constant  or  zero,  or 

uniformly  increasing  or  diminishing  with  the  time;  so  that  no  disturbance  depending 
on  J  can  be  propagated  as  a  wave. 

(100)  The  equations  of  the  electromagnetic  field,  deduced  from  purely  experimental 
evidence,  show  that  transversal  vibrations  only  can  be  propagated.  If  we  were  to  go 
beyond  our  experimental  knowledge  and  to  assign  a  definite  density  to  a  substance  which 


we  should  call  the  electric  fluid,  and  select  either  vitreous  or  resinous  electricity  as  the 
representative  of  that  fluid,  then  we  might  have  normal  vibrations  propagated  with  a 
velocity  depending  on  this  density.  We  have,  however,  no  evidence  as  to  the  density  of 
electricity,  as  we  do  not  even  know  whether  to  consider  vitreous  electricity  as  a  sub- 
stance or  as  the  absence  of  a  substance. 

Hence  electromagnetic  science  leads  to  exactly  the  same  conclusions  as  optical  science 
with  respect  to  the  direction  of  the  disturbances  which  can  be  propagated  through  the 
field;  both  affirm  the  propagation  of  transverse  vibrations,  and  both  give  the  same  velocity 
of  propagation.  On  the  other  hand,  both  sciences  are  at  a  loss  when  called  on  to  affirm 
or  deny  the  existence  of  normal  vibrations. 

Belation  between  the  Index  of  Refraction  and  the  Electromagnetic  Character  of  the 


(101)  The  velocity  of  light  in  a  medium,  according  to  the  Undulatory  Theory,  is 

i  V<>' 

where  i  is  the  index  of  refraction  and  V0  is  the  velocity  in  vacuum.  The  velocity, 
according  to  the  Electromagnetic  Theory,  is 

v    Awp. 
where,  by  equations  (49)  and  (71),  k=y.fr0,    and    k0=='4hrVl. 

Hence  D=-, (80) 


or  the  Specific  Inductive  Capacity  is  equal  to  the  square  of  the  index  of  refraction 

divided  by  the  coefficient  of  magnetic  induction. 

Propagation  of  Electromagnetic  Disturbances  in  a  Crystallized  Medium. 

(102)  Let  us  now  calculate  the  conditions  of  propagation  of  a  plane  wave  in  a 
medium  for  which  the  values  of  k  and  p  are  different  in  different  directions.     As  we. 
do  not  propose  to  give  a  complete  investigation  of  the  question  in  the  present  imperfect 
state  of  the  theory  as  extended  to  disturbances  of  short  period,  we  shall  assume  that  the 
axes  of  magnetic  induction  coincide  in  direction  with  those  of  electric  elasticity. 

(103)  Let  the  values  of  the  magnetic  coefficient  for  the  three  axes  be  X,  p,  v,  then 
the  equations  of  magnetic  force  (B)  become 




~  dy~ 




~  dz~ 








3  Y 





The  equations  of  electric  currents  (C)  remain  as  before. 
The  equations  of  electric  elasticity  (E)  will  be 


Q=4tt%,  1 (82) 

7   J 

where  47r#2,  4tt#2,  and  47rc2  are  the  values  of  k  for  the  axes  of  #,  y/,  z. 

Combining  these  equations  with  (A)  and  (D),  we  get  equations  of  the  form 

1  /  d2F        d*F      d*F\       \    d  (  d¥        dG  ,    dB.\      1  /rf2F     d*V\ 
r\Xdz*+Plf+Vdz*)-^fa{Xdx+f^^Vll)-7\W+^M)-         '      (bd) 

(104)  If  I,  m,  n  are  the  direction-cosines  of  the  wave,  and  V  its  velocity,  and  if 

lx-\-my-\-nz— Vt=tv, (84) 

then  F,  G,  H,  and  "V  will  be  functions  of  w ;  and  if  we  put  F,  G',  H',  Y'  for  the  second 
differentials  of  these  quantities  with  respect  to  w,  the  equations  will  be 

(vw  (£+£))  F'+  ^G'+^H'-JV*'=0, 



If  we  now  put 

V4  -  V2  —  \  l*},(  J> + &v)  +  w>(c2v + a2A) + n2^ + £» 

we  shall  find 

.  «2&2c2   //2   .  m2      n2\/79     ,       ,      ,      ,  ,      TT 


with  two  similar  equations  for  G'  and  H'.     Hence  either 



VF=N",  VG'=mf  and  VH'=n*'. 






The  third  supposition  indicates  that  the  resultant  of  F,  G',  H'  is  in  the  direction 
normal  to  the  plane  of  the  wave ;  but  the  equations  do  not  indicate  that  such  a  disturb- 
ance, if  possible,  could  be  propagated,  as  we  have  no  other  relation  between  ty'  and 
F,  G',  H'. 

The  solution  V  =  0  refers  to  a  case  in  which  there  is  no  propagation. 

The  solution  U  =  0  gives  two  values  for  V2  corresponding  to  values  of  F,  G',  H',  which 



are  given  by  the  equations 

-iF'  +  p2G'+-2H'=0, 

c  nv , 

(105)  The  velocities  along  the  axes  are  as  follows :- 

Direction  of  propagation 


Direction  of  the  electric  displacements    < 

















Xow  we  know  that  in  each  principal  plane  of  a  crystal  the  ray  polarized  in  that 
plane  obeys  the  ordinary  law  of  refraction,  and  therefore  its  velocity  is  the  same  in 
whatever  direction  in  that  plane  it  is  propagated. 

If  polarized  light  consists  of  electromagnetic  disturbances  in  which  the  electric  dis- 
placement is  in  the  plane  of  polarization,  then 

a2=bs=(f (93) 

If,  on  the  contrary,  the  electric  displacements  are  perpendicular  to  the  plane  of  pola- 

\=p=v (94) 

We  know,  from  the  magnetic  experiments  of  Fakaday,  Pluckek,  &c,  that  in  many 
crystals  X,  jt/,,  v  are  unequal. 

The  experiments  of  Knoblauch  *  on  electric  induction  through  crystals  seem  to  show 
that  a,  b  and  c,  may  be  different. 

The  inequality,  however,  of  X,  p,  v  is  so  small  that  great  magnetic  forces  are  required 
to  indicate  their  difference,  and  the  differences  do  not  seem  of  sufficient  magnitude  to 
account  for  the  double  refraction  of  the  crystals. 

On  the  other  hand,  experiments  on  electric  induction  are  liable  to  error  on  account 
of  minute  flaws,  or  portions  of  conducting  matter  in  the  crystal. 

Further  experiments  on  the  magnetic  and  dielectric  properties  of  crystals  are  required 
before  Ave  can  decide  whether  the  relation  of  these  bodies  to  magnetic  and  electric 
forces  is  the  same,  when  these  forces  are  permanent  as  when  they  are  alternating  with 
the  rapidity  of  the  vibrations  of  light. 

*  Philosophical  Magazine,  1852. 


Relation  between  Electric  Resistance  and  Transparency. 

(10G)  If  the  medium,  instead  of  being  a  perfect  insulator,  is  a  conductor  whose  resist- 
ance per  unit  of  volume  is  g>,  then  there  will  be  not  only  electric  displacements,  but  true 
currents  of  conduction  in  which  electrical  energy  is  transformed  into  heat,  and  the  undu- 
lation is  thereby  weakened.  To  determine  the  coefficient  of  absorption,  let  us  investi- 
gate the  propagation  along  the  axis  of  w  of  the  transverse  disturbance  G. 

By  the  former  equations 

=  -4^(f+?)  by(A)> 

d*G       ,    .       /l  d*G      ]  dG\  ,      _.        .  ,„  /n_. 

^  =  +^{kW-g-dt)hy(V)^\(F) (95) 

If  G  is  of  the  form 

G=.e-p*  cos  (qw+nt), (96) 

we  find  that 

JP=— C-=— ^-r, (97) 

§    g      §    * 
where  V  is  the  velocity  of  light  in  air,  and  i  is  the  index  of  refraction.     The  proportion 
of  incident  light  transmitted  through  the  thickness  x  is 

e--?x (98) 

Let  K  be  the  resistance  in  electromagnetic  measure  of  a  plate  of  the  substance  whose 
thickness  is  w,  breadth  b,  and  length  I,  then 

^H (99) 

(107)  Most  transparent  solid  bodies  are  good  insulators,  whereas  all  good  conductors 
are  very  opaque. 

Electrolytes  allow  a  current  to  pass  easily  and  yet  are  often  very  transparent.  We 
may  suppose,  however,  that  in  the  rapidly  alternating  vibrations  of  light,  the  electro- 
motive forces  act  for  so  short  a  time  that  they  are  unable  to  effect  a  complete  separation 
between  the  particles  in  combination,  so  that?  when  the  force  is  reversed  the  particles 
oscillate  into  their  former  position  without  loss  of  energy. 

Gold,  silver,  and  platinum  are  good  conductors,  and  yet  when  reduced  to  sufficiently 
thin  plates  they  allow  light  to  pass  through  them.  If  the  resistance  of  gold  is  the  same 
for  electromotive  forces  of  short  period  as  for  those  with  which  we  make  experiments, 
the  amount  of  light  which  passes  through  a  piece  of  gold-leaf,  of  which  the  resistance 
was  determined  by  Mr.  C.  Hockin,  would  be  only  10-50  of  the  incident  light,  a  totally 
imperceptible  quantity.     I  find  that  between  -g^y  and  ywoo  of  green  light  gets  through 


such  gold-leaf.  Much  of  this  is  transmitted  through  holes  and  cracks;  there  is  enough, 
however,  transmitted  through  the  gold  itself  to  give  a  strong  green  hue  to  the 
transmitted  light.  This  result  cannot  be  reconciled  with  the  electromagnetic  theory 
of  light,  unless  we  suppose  that  there  is  less  loss  of  energy  when  the  electromotive  forces 
are  reversed  with  the  rapidity  of  the  vibrations  of  light  than  when  they  act  for  sensible 
times,  as  in  our  experiments. 

Absolute  Values  of  the  Electromotive  and  Magnetic  Forces  called  into  play  in  the 

Propagation  of  Light. 

(108)  If  the  equation  of  propagation  of  light  is 


F  =  Acos:-(2-Va 
the  electromotive  force  will  be 


and  the  energy  per  unit  of  volume  will  be 



where  P  represents  the  greatest  value  of  the  electromotive  force.     Half  of  this  consists 
of  magnetic  and  half  of  electric  energy. 

The  energy  passing  through  a  unit  of  area  is 


so  that 

P  =X/8^VW, 

where  V  is  the  velocity  of  light,  and  W  is  the  energy  communicated  to  unit  of  area  by 
the  light  in  a  second. 

According  to  Pouillet's  data,  as  calculated  by  Professor  W.  Thomson*,  the  mecha- 
nical value  of  direct  sunlight  at  the  Earth  is 

83 -4  foot-pounds  per  second  per  square  foot. 
This  gives  the  maximum  value  of  P  in  direct  sunlight  at  the  Earth's  distance  from  the  Sun, 


or  about  600  Daxiell's  cells  per  metre. 

At  the  Sun's  surface  the  value  of  P  would  be  'about 

13,000  Daxiell's  cells  per  metre. 

At  the  Earth  the  maximum  magnetic  force  would  be  -193  f. 

At  the  Sun  it  would  be  4T3. 

These  electromotive  and  magnetic  forces  must  be  conceived  to  be  reversed  twice  in 
every  vibration  of  light ;  that  is,  more  than  a  thousand  million  million  times  in  a  second. 

*  Transactions  of  the  Royal  Society  of  Edinburgh,  1854  ("  Mechanical  Energies  of  the  Solar  System"). 
t  The  horizontal  magnetic  force  at  Kew  is  about  1-76  in  metrical  units. 



General  Methods. 

(109)  The  electromagnetic  relations  between  two  conducting  circuits,  A  and  B, 
depend  upon  a  function  M  of  their  form  and  relative  position,  as  has  been  already 

M  may  be  calculated  in  several  different  ways,  which  must  of  course  all  lead  to  the 
same  result. 

First  Method.  M  is  the  electromagnetic  momentum  of  the  circuit  B  when  A  carries 
a  unit  current,  or  „,    i  j  ,  v 


where  F,  G,  H  are  the  components  of  electromagnetic  momentnm  due  to  a  unit  current 
in  A,  and  ds'  is  an  element  of  length  of  B,  and  the  integration  is  performed  round  the 
circuit  of  B. 

To  find  F,  G,  Ff ,  we  observe  that  by  (B)  and  (C) 

with  corresponding  equations  for  G  and  H,  j?',  q',  and  r'  being  the  components  of  the 
current  in  A. 

Now  if  we  consider  only  a  single  element  ds  of  A,  we  shall  have 

p'=~ds,         q'  =  £ds,         r'=fsds, 

and  the  solution  of  the  equation  gives 

Y=^ds,         G=^fds,         K=^ds, 

§  as      '  g  as      '  §  as 

where  §  is  the  distance  of  any  point  from  ds.     Flence 


=  |  \  -  cos  6dsds', 

where  0  is  the  angle  between  the  directions  of  the  two  elements  ds,  ds',  and  §  is  the 
distance  between  them,  and  the  integration  is  performed  round  both  circuits. 

In  this  method  we  confine  our  attention  during  integration  to  the  two  linear  circuits 

(110)  Second  Method.  M  is  the  number  of  lines  of  magnetic  force  which  pass 
through  the  circuit  B  when  A  carries  a  unit  current,  or 


where  pa,  pfi,  py  are  the  components  of  magnetic  induction  due  to  unit  current  in  A, 


S'  is  a  surface  bounded  by  the  current  B,  and  I,  m,  n  are  the  direction-cosines  of  the 
normal  to  the  surface,  the  integration  being  extended  over  the  surface. 
We  may  express  this  in  the  form 

M=la,2  -o  sin  &  sin  §  sin  cpdS'ds, 

where  d&  is  an  element  of  the  surface  bounded  by  B,  ds  is  an  element  of  the  circuit  A, 
g  is  the  distance  between  them,  6  and  &  are  the  angles  between  g  and  ds  and  between 
g  and  the  normal  to  dS'  respectively,  and  <p  is  the  angle  between  the  planes  in  which 
6  and  &  are  measured.  The  integration  is  performed  round  the  circuit  A  and  over  the 
surface  bounded  by  B. 

This  method  is  most  convenient  in  the  case  of  circuits  lying  in  one  plane,  in  which 
case  sin  0=1,  and  sinp  =  l. 

111.  Third  Method.  M  is  that  part  of  the  intrinsic  magnetic  energy  of  the  whole 
field  which  depends  on  the  product  of  the  currents  in  the  two  circuits,  each  current 
being  unity. 

Let  a,  |3,  y  be  the  components  of  magnetic  intensity  at  a*ny  point  due  to  the  first 
circuit,  a',  j3',  y'  the  same  for  the  second  circuit ;  then  the  intrinsic  energy  of  the 
element  of  volume  dV  of  the  field  is 


The  part  which  depends  on  the  product  of  the  currents  is 

£-(««! +8p'+y>/)dY. 

Hence  if  we  know  the  magnetic  intensities  I  and  I'  due  to  unit  current  in  each  circuit, 
we  may  obtain  M  by  integrating 

J^lfj,!!' cos  MV 
Ait    ' 

over  all  space,  where  6  is  the  angle  between  the  directions  of  I  and  I'. 

Application  to  a  Coil. 

(112)  To  find  the  coefficient  (M)  of  mutual  induction  between  two  circular  linear 
conductors  in  parallel  planes,  the  distance  between  the  curves  being  everywhere  the  same, 
and  small  compared  with  the  radius  of  either. 

If  r  be  the  distance  between  the  curves,  and  a  the  radius  of  either,  then  when  r  is 
very  small  compared  with  a,  we  find  by  the  second  method,  as  a  first  approximation, 



To  approximate  more  closely  to  the  value  of  M,  let  a  and  ax  be  the  radii  of  the  circles, 
and  b  the  distance  between  their  planes ;  then 



We  obtain  M  by  considering  the  following  conditions : — 
1st.  M  must  fulfil  the  differential  equation 

d*M     rf2M     1  dM 

da?    '    dl)2    '  a   da 

This  equation  being  true  for  any  magnetic  field  symmetrical  with  respect  to  the  common 
axis  of  the  circles,  cannot  of  itself  lead  to  the  determination  of  M  as  a  function  of  a,  ax , 
and  J).     We  therefore  make  use  of  other  conditions. 

2ndly.  The  value  of  M  must  remain  the  same  when  a  and  ax  are  exchanged. 

3rdly.  The  first  two  terms  of  M  must  be  the  same  as  those  given  above. 

M  may  thus  be  expanded  in  the  following  series : — 

M-telo" -fl-J-1  a-^+  l  3*8+(fli-g)g_  ]  {W+{a-axy){a-a,)      &    1 
6   r  \    ~2      a     ~16  a"2  32  a3  ~       ') 

-±?:ak+  l  a~a'+  l  b2-S(a-a^)_  1  (66«-(fl-gl)«)(a-fl])      &    \ 
\         2      a     ~16  «2  48  «3  ^~       ')' 

(113)  We  may  apply  this  result  to  find  the  coefficient  of  self-induction  (L)  of  a  circular 
coil  of  wire  whose  section  is  small  compared  with  the  radius  of  the  circle. 

Let  the  section  of  the  coil  be  a  rectangle,  the  breadth  in  the  plane  of  the  circle  being 
c,  and  the  depth  perpendicular  to  the  plane  of  the  circle  being  b. 

Let  the  mean  radius  of  the  coil  be  «,  and  the  number  of  windings  n ;  then  we  find, 
by  integrating,  ^  -™ 

L— wA  \  \  \ Mfa#  x'y')dx  fy  dx'  <¥> 

where  M.(xy  x'y')  means  the  value  of  M  for  the  two  windings  whose  coordinates  are  xy 
and  w'y'  respectively ;  and  the  integration  is  performed  first  with  respect  to  x  and  y  over 
the  rectangular  section,  and  then  with  respect  to  x'  and  y'  over  the  same  space. 

L=47m2#<log6y+:j^— 3(0— ^)  cot  20— ^  cos  20— gCot201ogcos0— g  tan2  0  log  sin  0 1 

,  7rn2r2(.       8a/ri    .„_,,...,   „   ,  r   ,„„    .__        „.      n  _  /it       Asin30   ,   lcos40. 
+  ^l0Sy(2sm  ^+1)  +  3'45  +  27'475cOS  ^-3"2(2-^^^  +  5im^l0gC0S^ 

.  13  sin49.        .     .)    ,   0 
+  Ico?JlogSin^k 

Here  «=  mean  radius  of  the  coil. 
„      r=  diagonal  of  the  rectangular  section  =/V/#2+c2. 
„      0=  angle  between  r  and  the  plane  of  the  circle. 
„     n=  number  of  windings. 
The  logarithms  are  Napierian,  and  the  angles  are  in  circular  measure. 

In  the  experiments  made  by  the  Committee  of  the  British  Association  for  deter- 
mining a  standard  of  Electrical  Resistance,  a  double  coil  was  used,  consisting  of  two 
nearly  equal  coils  of  rectangular  section,  placed  parallel  to  each  other,  with  a  small 
interval  between  them. 


The  value  of  L  for  this  coil  was  found  in  the  following  way. 

The  value  of  L  was  calculated  by  the  preceding  formula  for  six  different  cases,  in 

which  the  rectangular  section  considered  has  always  the  same  breadth,  while  the  depth 


A,  B,  C,     A+B,     B+C,     A+B-f-C, 

and  n=l  in  each  case. 

Calling  the  results 

L(A),     L(B),     L(G),&c, 

we  calculate  the  coefficient  of  mutual  induction  M(AC)  of  the  two  coils  thus, 

2ACM(AC)=(A+B+C)2L(A+B+C)-(A+B)2L(A+B)-(B+C)2L(B  +  C)+B2L(B). 

Then  if  >?,  is  the  number  of  windings  in  the  coil  A  and  n2  in  the  coil  B,  the  coefficient 
of  self-induction  of  the  two  coils  together  is 


(114)  These  values  of  L  are  calculated  on  the  supposition  that  the  windings  of  the 
wire  are  evenly  distributed  so  as  to  fill  up  exactly  the  whole  section.  This,  however,  is 
not  the  case,  as  the  wire  is  generally  circular  and  covered  with  insulating  material. 
Hence  the  current  in  the  wire  is  more  concentrated  than  it  would  have  been  if  it  had 
been  distributed  uniformly  over  the  section,  and  the  currents  in  the  neighbouring  wires 
do  not  act  on  it  exactly  as  such  a  uniform  current  would  do. 

The  corrections  arising  from  these  considerations  may  be  expressed  as  numerical 
quantities,  by  which  we  must  multiply  the  length  of  the  wire,  and  they  are  the  same 
whatever  be  the  form  of  the  coil. 

Let  the  distance  between  each  wire  and  the  next,  on  the  supposition  that  they  are 
arranged  in  square  order,  be  D,  and  let  the  diameter  of  the  wire  be  d,  then  the  correc- 
tion for  diameter  of  wire  is 

+  2(log§  +  |log2  +  i-H). 

The  correction  for  the  eight  nearest  wires  is 

For  the  sixteen  in  the  next  row 


These  corrections  being  multiplied  by  the  length  of  wire  and  added  to  the  former 
result,  give  the  true  value  of  L,  considered  as  the  measure  of  the  potential  of  the  coil 
on  itself  for  unit  current  in  the  wire  when  that  current  has  been  established  for  some 
time,  and  is  uniformly  distributed  through  the  section  of  the  wire. 

(115)  But  at  the  commencement  of  a  current  and  during  its  variation  the  current  is 
not  uniform  throughout  the  section  of  the  wire,  because  the  inductive  action  between 
different  portions  of  the  current  tends  to  make  the  current  stronger  at  one  part  of  the 
section  than  at  another.     When  a  uniform  electromotive  force  P  arising  from  any  cause 



acts  on  a  cylindrical  wire  of  specific  resistance  §,  we  have 


where  F  is  got  from  the  equation 

d*F      1  d¥  , 

r  being  the  distance  from  the  axis  of  the  cylinder. 

Let  one  term  of  the  value  of  F  be  of  the  form  Trn,  where  T  is  a  function  of  the  time, 
then  the  term  of  p  which  produced  it  is  of  the  form 

Hence  if  we  write 

—  -^-n2Trn-\ 


/_,  dT\  (J.K   d*T     ,         /X7T,2       1  d3T      .  _ 

pz=  p+ a)  — 7  iwr—;\  w-& ^  -  &c- 

dt)       §    dt*'        §\  12.22  dt* 
The  total  counter  current  of  self-induction  at  any  point  is 

f/T_„W_IT  ,  ^^.£[!  -1-  ^V+&c 

from  £=0  to  £=oo 

When  (=0,  ,  =  0,        •••(§),  =P.     (^)p=0,&c. 

When  teco, .;,=  ?,     ...  (£)_=<>,     (?).=0,  &<=. 

n'n    /p       \    7    ...      1„     ,      1  u.r*  dT  ,      |*V        1        rf'T  .  ,    . 

from  t=0  to  =co 

/^T\  /  d2T\ 

When  t=0,    p  =  0  throughout  the  section,    .*.  (~^r)  =P?  (  ^2" )    =^?  &c- 

/dT\  /rf2T\ 

When  £=00,  p=0  throughout "*  I'd*)    =(^'  l-^2")     =0,  &c 

Also  if  I  be  the  length  of  the  wire,  and  R  its  resistance, 


and  if  C  be  the  current  when  established  in  the  wire,  C=  -jt- 

The  total  counter  current  may  be  written 



Now  if  the  current  instead  of  being  variable  from  the  centre  to  the  circumference  of 
the  section  of  the  wire  had  been  the  same  throughout,  the  value  of  F  would  have  been 



where  y  is  the  current  in  the  wire  at  any  instant,  and  the  total  countercurrent  would 

have  been 

f-f'lrfF.  I  _      3     I  L'C 


or  the  value  of  L  which  must  be  used  in  calculating  the  self-induction  of  a  wire  for 
variable  currents  is  less  than  that  which  is  deduced  from  the  supposition  of  the  current 
being  constant  throughout  the  section  of  the  wire  by  ^pl,  where  I  is  the  length  of  the 
wire,  and  p  is  the  coefficient  of  magnetic  induction  for  the  substance  of  the  wire. 

(116)  The  dimensions  of  the  coil  used  by  the  Committee  of  the  British  Association 
in  their  experiments  at  King's  College  in  1864  were  as  follows : — 

Mean  radius =#='158194 

Depth  of  each  coil =  b  =-01608 

Breadth  of  each  coil       ....  =  c  =  -01841 

Distance  between  the  coils .     .     .  =-02010 

Number  of  windings       .     .     .     .  %=313 

Diameter  of  wire =-00126 

The  value  of  L  derived  from  the  first  term  of  the  expression  is  437440  metres. 
The  correction  depending  on  the  radius  not  being  infinitely  great  compared  with  the 
section  of  the  coil  as  found  from  the  second  term  is  — 7345  metres. 

The  correction  depending   on  the  diameter  of  the  wire  is     ]        ...     » 

per  unit  oi  length J 

Correction  of  eight  neighbouring  wires -J--0236 

For  sixteen  wires  next  to  these +'0008 

Correction  for  variation  of  current  in  different  parts  of  section       —  -2500 

Total  correction  per  unit  of  length -22437 

Length 311-236  metres. 

Sum  of  corrections  of  this  kind 70        „ 

Final  value  of  L  by  calculation 430165      „ 

This  value  of  L  was  employed  in  reducing  the  observations,  according  to  the  method 
explained  in  the  Eeport  of  the  Committee*.  The  correction  depending  on  L  varies 
as  the  square  of  the  velocity.  The  results  of  sixteen  experiments  to  which  this  correc- 
tion had  been  applied,  and  in  which  the  velocity  varied  from  100  revolutions  in 
seventeen  seconds  to  100  in  seventy-seven  seconds,  were  compared  by  the  method  of 

*  British  Association  Reports,  1863,  p.  169. 


least  squares  to  determine  what  further  correction  depending  on  the  square  of  the 
velocity  should  be  applied  to  make  the  outstanding  errors  a  minimum. 

The  result  of  this  examination  showed  that  the  calculated  value  of  L  should  be 
multiplied  by  1*0618  to  obtain  the  value  of  L,  which  would  give  the  most  consistent 

We  have  therefore  L  by  calculation 430165  metres. 

Probable  value  of  L  by  method  of  least  squares 456748       „ 

Result  of  rough  experiment  with  the  Electric  Balance  (see  §  46)      410000        „ 

The  value  of  L  calculated  from  the  dimensions  of  the  coil  is  probably  much  more 
accurate  than  either  of  the  other  determinations.